THE 

STUDY  OF    ELECTRICITY 

BY  THE  DEDUCTIVE 

METHOD 


BY 


GEORGE  IRA  ALDEN 
B.S.,  M.M.VE. 


WORCESTER,    MASSACHUSETTS 

COMMONWEALTH    PRESS 

1919 


QC5/1? 


Copyright  1919 

BY  GEORGE  IRA  ALDEN 

All  Rights  Reserved 


PREFACE 


Many  years  ago  the  author  discovered  the  fact 
that  an  endless  flexible  shaft  revolving  about  its 
geometric  axis  was  a  helpful  analogy  of  the  transmis- 
sion of  energy  by  electricity.  The  applications  of 
this  analogy  have  been  extended  at  intervals,  and  a 
pamphlet  on  the  subject  was  published  in  1915.  . 

The  analogy  when  used  in  connection  with  a  single 
working  hypothesis  has  universal  application,  afford- 
ing the  student  the  advantage  of  the  deductive 
method  of  study.  A  considerable  number  of  students 
have  from  time  to  time  expressed  their  appreciation 
of  the  assistance  they  have  derived  from  the  use  of 
this  analogy. 

Only  quite  recently  has  the  attempt  been  made  to 
extend  the  analogy  and  its  accompanying  working 
hypothesis  to  the  solution  of  all  the  problems  of 
electric  transmission  of  energy  by  both  direct  and 
alternating  current.  The  result  has  led  to  the  publica- 
tion of  this  book. 

Caution — The  student  or  the  critical  reader  should 
keep  clearly  in  mind  that  the  analogy  used  as  a  basis 
for  deductive  study  is  not  here  presented  as  a  theory. 
Neither  the  value  nor  the  results  of  the  method  are 
at  all  dependent  upon  our  views  as  to  the  probability 
of  the  real  existence  of  the  elementary  flexible  shafts 
referred  to  in  the  analogy.  All  teachers,  I  believe, 

iii 


432164 


IV  PREFACE 

advocate  and  use  analogies.  What  we  claim  for  our 
analogy  is  its  universal  application  in  a  given  field — 
namely,  transmission  of  energy  by  electricity. 

I  take  pleasure  in  making  here  acknowledgment  of 
the  assistance  of  Mr.  Harry  B.  Lindsay,  who  arranged 
much  of  the  material  in  the  book. 


CONTENTS 

PAGE 

INTRODUCTION vii 

CHAPTER 

I     MAGNETISM       .     .    „     ;     ,     ,     .     .  1 

II     MECHANICS        .     .     .     .     .     .     ;     .  10 

III  STATEMENT  AND  APPLICATION  OF  ANAL- 

OGY AND  WORKING  HYPOTHESIS  .  14 

IV  APPLICATION  OF  ANALOGY  TO  THE  GEN- 

ERATOR    .     .     ....     .     .     .  27 

V    INDUCTION   .     %     .     ."    .     .     .     .     .  38 

VI    COMBINATION  OF  OUT  OF  PHASE  TORQUES  51 

VII    THE  CONDENSER    ....     .     .     .  69 

VIII     MEASURING  INSTRUMENTS     .     ....  78 

IX    GENERAL  REMARKS  AND  RECAPITULATION  86 

X    PROBLEMS 89 

APPENDIX  101 


INTRODUCTION 

The  various  branches  of  modern  science  have  been 
developed  primarily  and  mainly  by  the  use  of  the 
inductive  method  of  study.  When  the  deductive 
method  has  been  applicable,  it  has  usually  resulted 
in  a  great  saving  of  time  to  the  student,  and  in  added 
clearness  of  thinking.  When  the  deductive  method 
can  be  used,  the  student  proceeds  with  confidence  in 
the  method,  and  an  unusual  degree  of  satisf action  in 
the  results  obtained.  In  the  earlier  studies  in  all 
branches  of  science,  the  approach  was  doubtless  along 
the  line  of  the  inductive  method.  In  some  branches, 
like  chemistry,  there  seems  to  be  scarcely  any  other 
method  possible.  In  modern  astronomy,  or  in  ther- 
modynamics, on  the  contrary,  the  deductive  method 
has  wide  application.  It  is  clear  that  the  deductive 
method  requires  some  comprehensive  law,  principle, 
or  analogy,  as  a  basis  for  its  application.  Keppler 
discovered  the  laws  of  planetary  motion  mainly  by 
the  inductive  method.  By  repeated  and  continuous 
observations  made  by  himself,  and  by  the  study  of 
what  others  had  observed  and  plotted,  he  was  finally 
able  to  enunciate  the  laws  governing  the  motion  of  the 
planets,  which  are  known  as  Keppler's  Laws.  Later, 
Sir  Isaac  Newton  discovered  the  law  of  gravitation. 
With  this  law  as  a  basis,  the  mathematician  is  easily 
able  to  deduce  the  laws  of  Keppler.  The  inductive 
process  was  the  work  of  a  lifetime.  By  the  deductive 


viii  INTRODUCTION 

process,  the  mathematician  can  prove  the  laws  in  a 
day.  Astronomy  early  took  a  high  rank  among  the 
sciences.  It  held  this  rank  because  it  was  able  to 
predict  results.  The  astronomer  not  only  explains 
why  an  eclipse  occurs,  but  he  predicts  by  the  deductive 
method  the  exact  time  of  its  occurrence. 

In  some,  if  not  all  the  subjects  which  have  invited 
scientific  investigation,  there  have  been  at  times 
erroneous  or  inadequate  reasons  put  forward  to  ex- 
plain certain  phenomena.  Heat  was  at  one  time 
thought  to  be  a  material  substance  called  caloric. 
The  term  latent  heat,  when  first  used,  was  applied 
to  the  unexpected  appearance  of  heat,  the  cause  of 
which  was  not  really  known,  such,  for  instance,  as 
the  heat  developed  by  friction.  This  heat  was  said 
to  have  been  latent  in  a  body  until  called  forth  by 
some  peculiar  conditions;  but  the  relations  between 
the  conditions  which  resulted  in  the  heat,  and  the 
effect  produced,  were  not  understood.  When  it  was 
discovered  that  heat  was  a  mode  of  molecular  motion, 
the  term  latent  heat  was  retained,  but  given  an  en- 
tirely new  meaning. 

In  each  of  the  branches  of  science,  there  has  come  a 
time  when  the  discovery  of  a  new  truth  has  marked 
an  epoch  of  change,  and  of  important  progress  in  the 
subject.  In  chemistry  this  epoch  came  with  the 
recognition  of  the  indestructibility  of  matter.  So  long 
as  it  was  thought  that  material  might  be  destroyed 
in  the  process  of  an  experiment,  little  value  could  be 
attached  to  the  results.  The  acceptance  of  the  idea 
of  the  indestructibility  of  energy,  or  the  discovery  of 


INTRODUCTION  ix 

what  is  known  as  the  conservation  of  energy,  gave 
us  the  science  of  thermodynamics,  and  the  rapid 
development  in  the  use  of  steam  and  in  the  application 
of  its  energy  to  the  commercial  uses  of  modern 
civilization. 

When  we  come  to  the  subject  of  electricity,  we  find 
that  while  it  long  occupied  a  place  in  the  text  books 
of  physics,  it  had  little  or  no  commercial  value  until 
the  invention  of  the  dynamo,  or  electric  generator,  in 
about  1876.  When  it  was  found  that  mechanical 
energy  delivered  to  a  generator  could  be  transmitted 
with  little  difficulty  through  long  distances,  the  com- 
mercial value  of  this  mode  of  transmitting  energy 
brought  the  subject  of  electricity  into  prominence. 
In  the  case  of  electricity,  the  inductive  method  has 
been  entirely  successful  in  discovering  all  its  functions 
and  uses,  and  in  giving  to  the  world  the  full  benefit 
of  its  great  possibilities. 

In  preparing  this  volume  from  the  standpoint  of  a 
new"approach  to  the  study  of  electricity,  I  have  had 
in  mind,  primarily,  the  advantages  of  the  logical 
processes  of  thought,  the  economy  of  time,  and  inci- 
dentally, the  financial  as  well  as  the  educational  value 
of  the  deductive  method  of  study,  to  the  student  and 
the  engineer. 

I  have  no  doubt  that  teachers  as  well  as  students 
have  felt  the  lack  of  some  more  definite  guiding  prin- 
ciple relating  to  the  various  phenomena  of  that  seem- 
ingly elusive  and  mysterious  thing  which  we  call 
electricity.  Long  experience  in  the  use  of  electrical 
machinery,  and  in  the  handling  of  electric  power, 


x  INTRODUCTION 

brings  with  it  a  certain  confidence  as  to  what  to  do 
under  certain  circumstances;  what  various  mechan- 
isms can  accomplish,  and  how  they  should  be  used. 
But  this  confidence  is  much  more  readily  acquired 
when  the  experience  is  preceded  by  familiarity  with 
the  method  of  study  disclosed  in  this  volume. 

The  principles  and  laws  of  magnetism  and  magnetic 
circuits  have  been  assumed  to  be  acceptable  to  the 
student  in  the  form  in  which  they  are  stated  in  the 
usual  text  book.  Chapter  I  is  a  r£sum6  and  discus- 
sion of  the  essential  facts  of  magnetism,  stated  as 
briefly  as  is  consistent  with  the  necessity  for  a  clear 
understanding  of  the  subject  on  the  part  of  the 
student  who  is  about  to  take  up  the  study  of 
electricity. 

Attention  is  directed  to  the  Appendix,  wherein 
will  be  found  discussion  of  points  which,  for  the  sake 
of  clearness,  have  been  inserted  without  comment  in 
the  text. 

THE   STUDENT'S   GUIDE   TO   DEDUCTIVE   THOUGHT 

As  a  help  to  the  student,  there  is  placed  at  the 
beginning  of  each  chapter  a  brief  paragraph  in  which 
an  attempt  is  made  to  indicate  the  train  of  thought 
which  he  will  be  asked  to  follow  in  that  chapter. 


CHAPTER  I 

Magnetism — Magnets  and  magnetic  materials;  examples; 
magnetic  lines  of  force;  magnetic  field;  permeability.  Elec- 
tro-Magnetism— Electro-magnets  and  their  uses;  examples. 
Magnetic  Circuit — examples. 


STUDENT'S  GUIDE 

In  this  chapter  will  be  found  the  elementary  facts 
about  magnets  and  magnetism.  Magnetism  is  closely 
associated  with  all  electrical  phenomena,  and  it  is 
therefore  of  the  greatest  importance  to  study  this 
relation.  Simple  illustrations  have  been  chosen  for 
discussion  to  demonstrate  the  fundamental  principles 
involved. 

MAGNETISM 

Before  proceeding  to  the  study  of  electricity,  it  is 
essential  that  the  elementary  facts  of  magnetism  be 
reviewed.  A  magnet  may  exist  for  ages  without  the 
appearance  of  electricity,  but  no  energy  is  transmitted 
or  applied  by  electrical  means  without  the  appearance 
of,  and  usually  the  assistance  of  magnetism. 

It  was  early  observed  that  a  mineral  known  as 
magnetite  (FeaO^  had,  in  its  natural  state,  the  prop- 
erty of  attracting  to  itself  and  holding  pieces  of  iron 
and  steel.  This  property  of  attraction  is  called 
magnetism,  and  the  body  possessing  it  is  called  a 
magnet.  A  piece  of  iron  ore  which  possesses  inherent 
magnetism  is  called  a  natural  magnet.  A  piece  of  steel 


^J  V    I        °Ji       :  /u     .CHAPTER   I 

which  has  been  magnetized  is  called  an  artificial 
magnet.  There  are  other  substances  capable  of 
being  attracted  by  a  magnet,  but  their  susceptibility 
to  magnetism  is  small  and  relatively  of  little  import- 
ance; cobalt,  nickel,  chromium,  and  manganese  are 
such  materials. 

The  most  common  example  of  a  magnet  is  the 
ordinary  woodsman's  or  mariner's  compass  needle, 
which  is  simply  a  small  magnet  pivoted  and  mounted 
so  that  when  undisturbed  by  local  magnetic  influences 
it  will  indicate  magnetic  north  and  south.  That  end 
of  the  needle  which  always  points  north  is  called  the 
north  pole  of  the  magnet,  and  the  other  end  is  called 
the  south  pole.  This  action  of  the  compass  needle  is 
due  to  the  fact  that  the  earth  itself  exhibits  magnetic 
polarity,  there  being  a  magnetic  pole  in  the  vicinity  of 
each  of  the  geographic  poles. 

To  explain  the  influence  of  one  magnet  on  another, 
or  of  a  magnet  on  a  piece  of  steel,  the  existence  of  lines 
of  magnetic  force  has  been  assumed;  and  the  region 
about  a  magnet  where  magnetic  effects  may  be  de- 
tected is  called  a  field  of  force  or  magnetic  field,  and 
is  assumed  to  be  traversed  by  a  greater  or  lesser  num- 
ber of  lines  of  force  according  to  the  strength  of  the 
magnet.  (See  Fig.  1.)  It  will  be  seen  a  little  later 
that  magnetic  lines  of  force  always  seem  to  exist  in  the 
presence  of  an  electric  current,  the  magnetic  field  thus 
produced  being  similar  in  properties  to  that  existing 
around  a  magnet. 

It  is  convenient  for  discussion,  and  necessary  for  the 
explanation  of  certain  electrical  phenomena,  to  state 


MAGNETISM  3 

a  few  of  the  conventions  commonly  applied  to  mag- 
netic lines  and  fields  of  force. 

Lines  of  force  (sometimes  called  magnetic  flux) 
emerge  from  north  magnetic  poles  and  enter  south 
poles. 

Lines  of  force  are  closed  loops,  tending  always  to 
become  shorter,  having  a  repellant  effect  on  each 
other  and  being  established  with  much  greater  ease 
in  steel  and  iron  than  in  any  other  substance. 


FIG.  1 

The  repellant  effect  is  assumed  to  explain  the  notice- 
able repulsion  of  like  poles;  and  the  tendency  to 
become  shorter,  combined  with  the  ease  of  the  estab- 
lishment of  lines  in  iron  and  steel,  is  an  assumption 
made  in  order  to  explain  the  attractive  force  exerted 
by  a  magnet  on  a  bit  of  iron  or  steel. 

The  property  of  iron  and  steel  which  promotes  the 
apparent  strength  of  lines  of  force  is  called  permeabil- 
ity; thus  of  two  bars  of  iron  subjected  to  the  same 
magnetizing  force,  the  one  which  appears  to  sustain 
the  greater  number  of  lines  of  force  is  said  to  have  the 


4  CHAPTER   I 

higher  permeability;  or  expressed  negatively,  to  have 
the  lower  reluctance.  In  general,  the  softer  the  iron, 
the  higher  the  permeability. 

Iron  and  steel  differ  also  to  a  great  degree  in  reten- 
tiveness  of  magnetism ;  soft  iron  loses  nearly  all  of  its 


FIG.  2 


magnetism  when  the  magnetizing  force  is  removed, 
whereas  steel  retains  a  considerable  proportion  of  its 
magnetism  indefinitely. 

The  magnetism  evidenced  whenever  an  electric 
current  is  established,  and  which  persists  while  the 
current  is  flowing,  but  disappears  when  the  current 
ceases  to  flow,  is  called  electro-magnetism. 


MAGNETISM 


To  be  assured  of  the  presence  of  a  magnetic  field 
in  the  space  about  a  conductor  carrying  an  electric 
current,  we  have  only  to  pass  a  compass  needle  over 
or  under  the  conductor.  (See  Fig.  2.)  The  violent 


Posmve 


CURRENT 


FIG.  3 


FIG.  4 

deflection  of  the  needle  is  visible  evidence  of  the 
action  of  a  force,  and  simple  experiments  with  the 
compass  needle  and  a  straight  conductor  carrying  a 
current  have  led  to  the  following  common  assump- 
tions : 


6  CHAPTER  I 

The  lines  of  force  are  circles  concentric  with  the 
conductor  (Fig.  3).  Their  direction  is  clockwise  when 
viewed  so  that  the  current  is  in  the  positive  direction 
(Figs.  3  and  4). 

Quantitative  measurements  show  that  in  air  or  in 
any  non-magnetic  material  the  distance  from  the  wire 
at  which  force  may  be  detected  is  directly  propor- 
tional to  the  strength  of  the  current. 

If  a  wire  be  wound  into  a  coil,  the  lines  of  force 
produced  by  one  turn  act  in  the  same  direction  as 
those  of  the  adjoining  turns,  and  in  this  manner  a 
field  of  force  of  considerable  strength  may  be  estab- 
lished by  a  very  moderate  current.  By  this  means 
magnets  called  electro-magnets  may  be  made,  which 
are  dependent  for  magnetism  upon  the  current  in  the 
coils  of  wire  with  which  they  are  wound.  When 
arranged  in  proper  mechanical  form,  a  magnet  may  be 
attached  to  great  loads  of  metal  simply  by  passing 
current  through  the  magnet  coils,  and  the  release  of 
the  material  lifted  is  accomplished  by  stopping  the 
flow  of  current. 

Electro-magnets  form  the  actuating  force  of  a  great 
number  of  electrical  devices,  such  as  telegraph  sound- 
ers, annunciators,  door  openers,  relays,  circuit- 
breakers,  watchman  clocks,  and  electric  devices  of 
many  sorts. 

It  is  customary  to  speak  of  the  path  of  the  lines  of 
force  as  the  magnetic  circuit ;  and  where  this  path  is  in 
iron  or  steel  throughout  most  of  its  length,  no  atten- 
tion is  given  to  the  space  outside  the  metal  which 
constitutes  the  core  of  the  magnet,  and  calculations 


MAGNETISM 


are  made  as  if  all  of  the  magnetic  flux  traversed  the 
core. 

Figure  5  shows  the  magnetic  circuit  of  an  ordinary 
electric  bell.  A,  B,  C,  and  D  are  made  of  soft  iron. 
The  coils  of  wire  are  slipped  on  over  the  magnet  cores 


~e* 


' 8 


FIG.  5 

A  and  C  before  the  armature  D  is  put  in  place.  F  is  a 
spring  support  for  the  armature,  which  allows  it  to 
vibrate;  £  is  a  spring  contact  through  which  current 
is  led  to  supply  the  magnet  coils.  The  dotted  line 
shows  the  path  of  the  magnetic  lines.  The  operation 
of  the  bell  is  as  follows:  current  flows  through  the 
wire  wound  around  the  magnet  cores  Cand  A,  through 


CHAPTER  I 


FIG.  6 


YOKE 


POLS  SHOS 


POLE 


FIG.  7 


MAGNETISM  9 

the  spring  F,  armature  D,  and  out  at  E.  The  mag- 
netic attraction  of  A  and  C  for  D  causes  the  armature 
to  draw  E  away  from  the  fixed  contact  G,  breaking 
the  current.  As  soon  as  the  current  is  broken,  the 
magnetism  is  destroyed  and  spring  F  returns  D  to 
the  position  shown  in  Fig.  5,  when  the  action  is  re- 
peated, giving  a  rapid  vibratory  motion  to  the  arma- 
ture which  carries  the  bell  tapper. 

Figures  6  and  7  show  the  magnetic  circuits  of  a 
transformer  and  a  four-pole  generator  respectively. 
The  dotted  lines  represent  the  path  of  the  lines  of 
magnetic  force.  These  pieces  of  apparatus  are  dis- 
cussed in  later  chapters  with  respect  to  their  electrical 
characteristics,  and  are  introduced  here  simply  to 
show  the  magnetic  circuits  of  typical  cases. 


CHAPTER  II 

Mechanics — Definitions;  conservation  of  energy;  power  trans- 
mitted by  a  shaft ;  example  involving  power  and  energy. 


STUDENT'S  GUIDE 

In  this  chapter  the  fundamental  mechanical 
quantities  are  stated  and  defined.  These  are  the 
basic  definitions  which  we  must  have  well  in  mind 
before  we  can  proceed  to  the  study  of  electrical 
transmission  of  energy  with  confidence  in  our  ability 
to  apply  logical  tests  to  the  conclusions  at  which  we 
wish  to  arrive. 

MECHANICS 

In  order  that  we  may  profitably  approach  the  sub- 
ject of  electrical  transmission  of  energy,  we  must 
first  review  the  principles  of  mechanics,  stating 
definitions  and  recording  certain  formulae  for  future 
reference. 

DEFINITIONS 

Mass  is  the  quantity  of  matter  of  which  a  body  is 
composed. 

Motion  is  a  change  of  position  of  a  body  relative 
to  some  point  of  reference. 

Force  is  that  which  produces,  tends  to  produce,  or 
to  modify  motion. 

Weight.  Weight  is  the  force  exerted  on  a  body  by 
virtue  of  the  attraction  of  gravity. 


MECHANICS  11 

Velocity  or  Speed  is  the  rate  of  motion  or  rate  of 
change  of  position.  Velocity  is  measured  in  feet  per 
second,  miles  per  hour,  revolutions  per  minute,  etc. 

Acceleration  is  the  rate  of  change  of  velocity.  It  is 
measured  in  feet  per  second  per  second,  miles  per 
hour  per  minute,  degrees  per  second  per  second, 
etc.  The  application  of  force  to  a  mass  whose 
motion  is  restricted  only  by  its  inertia  causes  accel- 
eration, and  the  product  of  the  mass  times  the 
acceleration  equals  the  force  exerted.  F  =  MA. 

Angular  Velocity.  When  a  body  rotates,  its  parti- 
cles move  in  circles  about  some  line  in  the  body 
called  the  axis  of  rotation.  The  rate  of  motion  of  a 
particle  along  the  circumference  of  one  of  these  circles, 
expressed  in  degrees  per  unit  of  time,  is  called  angular 
velocity.  Angular  velocity  is  numerically  equal  to 
the  linear  velocity  of  a  point  at  unit  radius  from  the 
axis  of  rotation. 

Energy,  Work,  Power.  Work  is  done  when  a  force 
causes  motion,  and  is  equal  to  the  product  of  the 
force  and  the  distance  through  which  it  acts.  Energy 
is  capacity  for  doing  work.  Power  is  the  rate  of 
application  of  energy,  or  the  rate  of  doing  work. 

Energy  and  power  are  frequently  confused;  they  are 

£ 
related  thus,  P  =  p  where  P  =  power,  E  =   energy, 

and    T    =    time.     One  horsepower    =    33,000   foot 
pounds  per  minute  =  746  watts. 

Stress  and  Strain.  When  force  is  applied  to  a  body, 
whether  motion  results  or  not,  the  body  is  under 
stress;  the  resulting  deformation  is  called  strain. 


12  CHAPTER   II 

Stress  which  tends  to  lengthen  the  object  to  which 
it  is  applied  is  called  tension;  stress  which  tends  to 
shorten  or  squeeze  the  object  is  called  compression; 
stress  which  tends  to  cut  an  object  by  moving  one 
section  across  another  is  called  shear;  and  stress 
which  tends  to  twist  an  object  is  called  torsion. 

Rigidity,  Flexibility.  A  body  which  resists  de- 
formation is  said  to  be  rigid,  or  to  possess  rigidity;  a 
body  which  is  easily  bent  is  called  flexible.  It  is  to 
be  noted  that  a  body  may  be  rigid  with  regard  to  one 
sort  of  stress,  and  flexible  to  another;  for  example,  a 
flat  piece  of  steel  is  rigid  under  tension,  but  flexible  in 
bending. 

Elasticity.  A  body  which  when  strained  returns 
completely  or  nearly  to  its  unstressed  dimensions,  is 
said  to  be  elastic.  Most  materials  have  a  definite 
stress  beyond  which  they  lose  their  elasticity.  When 
this  elastic  limit  is  reached,  the  material  loses  much 
of  its  rigidity.  A  non-elastic  body  is  called  plastic. 
Metals  become  plastic  when  strained  beyond  their 
elastic  limit. 

Torque.  The  torque  or  moment  of  a  force  about 
any  axis  is  equal  to  the  product  of  the  force  and  its 
perpendicular  distance  from  the  axis. 

Inertia.  Inertia  is  that  property  of  matter  by 
virtue  of  which  it  resists  change  in  its  state  of  rest 
or  motion. 

The  Conservation  of  Energy.  An  understanding  of 
the  principle  of  the  conservation  of  energy  is  most 
important.  Logic  and  experience  indicate  that  the 
quantity  of  energy  in  the  universe  is  constant,  and  the 


MECHANICS  13 

energy  which  can  be  transferred  from  a  body  or  a 
system  of  bodies  is  limited.  Energy  expended  as 
work,  is  transformed  into  a  different  form;  no  energy 
is  destroyed  or  created. 

Power  Transmitted  by  a  Rotating  Shaft.  If  a  force 
P  is  exerted  at  a  radius  R  from  the  axis  of  rotation  of 
a  shaft  which  makes  N  revolutions  per  minute,  the 
torque  exerted  is  PR,  the  distance  traveled  by  the 
force  is  2^^  per  minute.  The  work  done  per 
revolution  is  2irRP,  and  per  minute  is  2irRNP. 
2irN  is  the  angular  velocity  of  the  shaft  in  degrees  per 
minute  (IT  =  180°,  in  this  sense),  and  PR  is  torque. 

2irRNP  =  (PR)  (2-n-N)  =  MA  =  power,  where 
A  =  2irN  =  angular  velocity,  and  M  =  PR  = 
moment  or  torque. 

This  last  expression  should  be  thoroughly  under- 
stood and  memorized. 

Power  and  energy  should  be  as  distinctly  separate 
in  the  student's  mind  as  speed  and  distance.  For 
example,  if  a  man  walk  at  the  rate  of  4  miles  per 
hour  for  3  hours,  he  will  travel  12  miles.  Rate  X  time 
=  distance.  If  a  man  is  buying  energy  at  the  rate 
of  75  horsepower  and  he  uses  energy  at  that  rate 
for  8  hours,  he  has  used  600  horsepower  hours;  or 
since  one  horsepower  is  equal  to  33,000  foot  pounds 
per  minute,  75  X  33,000  =  the  power  in  foot  pounds 
per  minute.  Eight  hours  equal  480  minutes;  there- 
fore, 75  X  33,000  X  480  =  1,188,000,000  foot  pounds 
of  energy  used.  Here,  as  before,  power  X  time  = 
energy. 


CHAPTER  III 

Electricity  Defined.  The  Analogy  Stated — The  electric  circuit; 
statement  of  working  hypothesis.  Algebraic  Expression  for 
Rate  of  Transmission  of  Energy.  Deduction  of  Ohm's  Law. 
Current  Direction  Defined — Alternating  current.  Analogy 
Applied  to  Particular  Circuit.  Electric  Batteries — Primary 
cells;  storage  cells. 

STUDENT'S  GUIDE 

Having  covered  the  preliminary  ground,  we  shall 
now  proceed  to  the  statement  of  the  analogy  on 
which  the  work  is  based.  Then,  with  the  outlin- 
ing of  a  working  hypothesis,  we  are  equipped  for  the 
investigation  of  any  electrical  phenomenon.  The  first 
subject  treated  is  Ohm's  Law,  after  which  a  simple 
electric  circuit  is  discussed  in  terms  of  the  analogy. 
Electric  batteries  are  described,  with  notes  on  the 
charging  of  storage  cells. 

The  function  of  electricity  is  to  transmit  energy. 
This,  so  far  as  we  are  aware,  is  its  only  function. 
The  energy  is  transmitted  by  means  of  a  conductor, 
commonly  a  copper  wire.  In  practice,  energy  is 
transmitted  in  this  way  several  hundred  miles  with 
success,  and  with  comparatively  small  losses  due  to 
transmission.  We  define  electricity  from  its  single 
function,  and  we  say  that,  for  our  purpose,  electricity 
is  a  method  of  transmitting  energy.  We  need  go  no 
farther  than  this  for  a  definition.  We  need  not 


WORKING   HYPOTHESIS  15 

assume  that  there  is  any  such  entity  as  electricity. 
It  is  just  a  method  of  doing  something.  We  send 
energy  a  long  distance  by  electricity,  and  we  send 
parcels  a  long  distance  by  express.  The  question  is 
naturally  raised,  "  But  how  is  the  energy  transmitted? 
If  we  do  not  know  exactly  how  it  is  transmitted,  is 
the  transmission  like  anything  we  do  know  about?" 

Our  answer  to  this  question  is  in  the  affirmative, 
and  we  answer  it  by  saying  that  THE  TRANSMIS- 
SION OF  ENERGY  BY  ELECTRICITY  IS  LIKE 
THE  TRANSMISSION  OF  ENERGY  BY  AN 
ENDLESS  SHAFT  OF  NEGLIGIBLE  MASS, 
PERFECTLY  FLEXIBLE  WITH  REGARD  TO 
BENDING,  BUT  REASONABLY  RIGID  AND 
ELASTIC  WITH  REGARD  TO  TORSION  AND 
REVOLVING  ABOUT  ITS  GEOMETRIC  AXIS. 
This  is  our  comprehensive  analogy  for  the  trans- 
mission of  energy  by  electricity.  Following  out  this 
analogy,  if  we  inquire  what  is  the  formula  for  the 
transmission  of  energy  by  shafting,  the  answer, 
known  to  every  student  familiar  with  the  rudiments 
of  mechanics,  is  this:  "The  rate  of  transmission  of 
energy  by  shafting  is  the  moment  or  torque  on  the 
shaft,  multiplied  by  its  angular  velocity  of  rotation." 

Now  we  may  picture  in  our  minds  a  conductor  of 
electricity  as  composed  of  an  indefinite  number  of 
infinitesimal  flexible  shafts,  each  capable  of  rotation 
about  its  own  geometric  axis.  We  may  consider  each 
of  these  elementary  flexible  shafts  as  made  up  of  a 
line  of  molecules  of  the  conductor.  A  good  con- 
ductor is  one  whose  molecules  are  of  such  shape  and 


16  CHAPTER   III 

character  as  to  enable  these  elementary  shafts  to  re- 
volve without  undue  interference  with  one  another. 
A  poor  conductor  of  electricity  may  be  considered  as 
a  body  whose  molecules  are  of  such  shape  and  char- 
acter, or  have  such  relations  to  one  another,  that 
these  lines  of  molecules  mutually  interfere  and  thus 
present  resistance  to  rotation. 

An  electric  circuit  consists  of  a  continuous  con- 
ductor which  is  capable  of  sustaining  an  electric 
current.  The  circuit  is  always  a  closed  curve,  and 
ordinarily  includes  the  following:  a  source  of  elec- 
tromotive force,  a  load,  and  a  wire  extending  from 
one  terminal  of  the  source  to  the  point  of  applica- 
tion of  the  energy  (at  the  load)  and  back  to  the 
other  terminal.  Obviously,  the  source  of  the  e.m.f. 
and  the  load  itself  are  parts  of  the  continuous  path 
which  constitutes  the  circuit.  An  "open  circuit" 
occurs  when  the  conductor  is  cut,  or  when  a  por- 
tion of  it  is  replaced  by  a  non-conductor;  when  the 
circuit  is  complete,  it  is  referred  to  as  a  "closed 
circuit."  It  should  be  observed  that  the  presence 
of  current  is  always  proof  that  a  closed  circuit 
exists,  but  that  a  circuit  may  be  closed  without  the 
inclusion  of  a  source  of  electromotive  force,  in  which 
case  no  current  will  be  present. 

While  it  is  unnecessary,  as  it  is  impossible,  to 
answer  all  the  questions  that  may  be  asked  in  con- 
nection with  our  analogy,  such  as  how  the  molecules 
may  connect  themselves  with  one  another,  etc.,  it  is 
helpful  for  the  application  of  this  analogy  to  the 
complicated  problems  arising  in  connection  with 


WORKING   HYPOTHESIS  17 

alternating  currents,  to  inquire  what  we  may  consider 
as  taking  place  in  the  magnetic  field  of  the  electric 
generator.  We  know  that  when  a  coil  of  copper  wire 
is  moved  in  the  magnetic  field,  a  current  is  set  up  in 
the  coil.  Here  we  supplement  our  analogy  by  a 
working  hypothesis  as  to  the  action  which  takes 
place  in  the  coil  under  these  circumstances.  This 
working  hypothesis  is  as  follows: 

WHEN  A  CONDUCTOR  IN  A  CLOSED  CIR- 
CUIT IS  MOVED  IN  A  MAGNETIC  FIELD, 
THE  MOTION  TENDS  TO  CONSTANTLY 
MAGNETIZE  THE  MOLECULES  OF  THE 
CONDUCTOR,  THE  INSTANTANEOUS  POLAR 
AXES  OF  THE  MOLECULES  BEING  AT  RIGHT 
ANGLES  TO  THE  LINES  OF  FORCE  IN  THE 
MAGNETIC  FIELD,  AND  THE  NORTH  POLES 
IN  THE  GENERAL  DIRECTION  OF  THE  MO- 
TION OF  THE  CONDUCTOR  RELATIVELY 
TO  THE  LINES  OF  FORCE  IN  THE  FIELD. 

We  find  that,  following  this  working  hypothesis  in 
connection  with  our  analogy,  we  are  led  to  results  that 
explain  and  agree  with  all  of  the  principal  phenomena 
connected  with  the  electrical  transmission  of  energy, 
either  by  direct  or  alternating  current,  including 
induction,  self-induction,  lag,  lead,  capacity,  etc.  We 
therefore  present  the  application  of  this  working 
hypothesis  in  some  detail. 

If  the  transmission  of  energy  by  electricity  is  like 
transmission  by  innumerable  tiny  flexible  shafts 
revolving  at  high  velocity,  we  can  readily  write  the 


18  CHAPTER    III 

algebraic  expression  for  the  rate  at  which   energy 
is  thus  being  transmitted. 

Let  A  =  the  common  angular  velocity  of  the 
flexible  shafts. 

Let  M  —  the  combined  torque  on  all  the  shafts. 

Then  it  is  well  known  and  easily  demonstrated  that 
the  rate  at  which  energy  (W)  is  being  transmitted  is 

Rate  of  transmission  =  A  X  M  Equation  1 

Ohm's  Law  is  expressed  by  the  formula 

E  =  C  X  R  Equation  2 

in  which 

E  =  electromotive  force  (volts); 

C  —  current  (amperes); 
and 

R  =  resistance  of  the  conductor. 

Multiplying  both  members  of  Equation  2  by  C 
gives 

CE  =  OR  Equation  3 

But 

CE  =  rate  of  transmission  by  electric  current; 
and 

AM  —  rate  of  transmission  by  flexible  shafts. 

The  two  expressions  CE  and  AM  are  identical  in 
form,  and  each  represents  rate  at  which  energy  is 
being  transmitted.  Therefore,  C  in  Equation  3  must 
be  angular  velocity,  and  E  must  be  torque.  Thus 
our  analogy  leads  us  to  the  correct  expression  for 
rate  of  electric  transmission  and  also  enables  us  to 
define  electromotive  force  as  torque,  and  current  as 
angular  velocity. 

Making  C  =  1  in  Equation  3,    we  have  E  =  R. 


WORKING  HYPOTHESIS  19 

Thus,  R  is  the  e.m.f.  when  C  is  unity.  E  is  also  the 
energy  when  C  =  1 ;  (E  X  1  =  E  units  of  energy). 

The  power  equation,  P  =  CE,  lends  itself  readily 
to  deductive  analysis,  and  mechanical  considerations 
lead  from  it  to  the  deduction  of  Ohm's  Law. 

If  we  take  a  simple  closed  circuit,  in  which  a  current 
is  flowing,  and  examine  it  to  determine  the  factors 
upon  which  the  rate  of  application  of  energy  depends, 
we  observe  that  a  certain  resistance  to  rotation  is 


FIG.  8 

encountered  and  overcome  between  the  molecular 
filaments,  the  energy  in  this  case  being  dissipated 
as  heat. 

This  resistance  is  developed  between  bodies  rotat- 
ing in  the  same  direction  causing  resistance  at  their 
lines  of  contact.  In  Fig.  8,  X  and  Y  represent  cross- 
sections  of  molecular  filaments  of  a  body  in  which 
current  is  flowing.  The  doubling  of  the  angular 
velocity  of  either  filament  will  double  the  power 
required  to  overcome  the  resistance,  and  the  doub- 
ling of  the  velocity  of  both  will  quadruple  the  power. 
Hence  the  power  required  to  overcome  resistance 
is  proportional  to  the  square  of  the  angular  velocity 
of  rotation  of  the  filaments,  and  the  general  form  of 
the  power  equation  (see  Chapter  III)  becomes 

Power  =  MA  =  A*K 
or  in  electrical  units 

EC  =  CR 


20  CHAPTER   III 

where  R  is  a  constant  depending  on  the  material 
and  size  of  the  conductor. 

By  dividing  both  sides  of  this  expression  by  C  we 
get  E  =  CR,  which  is  the  familiar  form  of  Ohm's  Law. 

In  view  of  the  foregoing,  the  so  called  "flow"  of 
an  electric  current  becomes  an  easily  understandable 
mechanical  operation  involving  a  simple  form  of 
motion — namely,  rotation.  The  direction  of  current 
flow  is  a  mere  convention,  and  we  may  choose  one 
direction  of  rotation  of  our  molecular  filaments  as 


/\  /\  /\  /\  /\  y\  A  A  /\  /v  A  A  A  /\  A  A  /\.  /\  /\  .^V  /\  /\ 

^-^  fC 


FIG.  9 

corresponding  to  the  positive  direction  of  current 
flow,  and  the  reverse  for  negative  current.  If  the 
applied  torque  is  oscillating,  then  the  motion  accom- 
panying the  electric  transmission  of  energy  will  be 
oscillating  rotary  motion,  and  the  current  is  said 
to  be  alternating. 

As  stated  above,  the  direction  of  rotation  of  the 
molecular  filaments  could  be  chosen  arbitrarily, 
but  the  logical  application  of  the  working  hypothesis 
stated  in  this  chapter  leads  (see  Fig.  10)  to  the  con- 
clusion that  the  direction  of  rotation  of  the  filaments 
and  the  direction  of  the  current  are  related  to  each 


WORKING   HYPOTHESIS  21 

other  as  the  direction  of  rotation  and  forward  travel 
of  an  ordinary  corkscrew. 

Let  us  now  examine  a  simple  electric  circuit,  first 
in  the  nomenclature  of  the  physicist,  in  terms  of 
potential  difference,  amperes,  and  ohms,  and  second 
in  the  light  of  our  analogy. 

We  shall  assume  that  G  is  a  direct-current  generator 
impressing  100  volts  on  AB,  a  non-inductive  resistance 
of  10  ohms.  V  is  a  voltmeter  which  requires  only  a 
negligible  current  to  give  a  reading  of  100  when 
placed  across  AB.  If  the  leads  GA  and  GB  are  good 
conductors  and  of  large  cross-section,  the  voltmeter 
will  also  read  100  when  applied  to  the  terminals  of 
the  generator. 

The  ammeter  A'  will  read  10  amperes. 

If  AB  is  a  uniform  wire  or  rod,  and  C  is  its  middle, 
the  voltmeter  in  the  position  shown  will  register  50 
volts. 

The  energy  expended  will  be  El  =  PR  =  1,000 
watts,  and  in  the  example  shown  will  be  entirely 
absorbed  in  heating  the  resistance  AB.  With  the 
polarity  shown,  the  current  will  be  flowing  from  the 
generator  terminal  to  B,  to  A,  and  back  to  G. 

As  C  is  moved  nearer  A,  the  reading  of  the  volt- 
meter will  become  smaller  and  smaller,  reading  25 
when  three-quarters  of  the  distance  from  B  to  A, 
reading  10  when  nine- tenths  of  the  distance,  these 
readings  being  in  accordance  with  the  law  of  the 
"fall  of  potential"  around  a  circuit. 

Let  us  take  the  above  statements  and  translate 
them  into  the  language  of  our  analogy. 


22  CHAPTER   III 

The  generator  applies  a  torque  to  the  ends  .of  the 
circuit  GBA  in  such  a  manner  as  to  cause  the  molecu- 
lar filaments  of  the  circuit  to  rotate,  this  rotation 
being  opposed  by  a  resistance  akin  to  friction  in  the 
portion  BA,  of  such  magnitude  that  heat  is  pro- 
duced. The  torque  indicator  (voltmeter)  is  so 
sensitive  that  no  appreciable  energy  is  required  for  its 
indication.  (See  Chapter  IX.) 

If  the  leads  GA  and  GB  are  short  and  of  large 
cross-section,  the  energy  loss  by  resistance  in  them 
will  be  so  small  as  to  make  no  difference  in  the  torque 
indication  whether  taken  across  AB  or  directly  at  the 
generator  terminals. 

The  reading  of  the  current  meter  is  a  number,  and 
with  the  idea  of  angular  velocity  in  mind  we  shall, 
for  the  present,  accept  that  number  as  being  simply 
a  cardinal  factor  in  the  energy  equation. 

If  the  torque  indicator  reads  100  across  AB,  it  will 
read  50  in  the  position  shown  in  the  sketch,  for  the 
angular  stress  on  a  molecular  fiber  will  be  one-half 
as  great  between  the  middle  and  either  end  as  between 
the  two  ends,  because  the  torque  of  resistance  is 
exerted  uniformly  throughout  the  length  of  each 
fiber.  This  proportionality  will  hold  for  fractional 
points  along  the  wire  or  rod. 

The  leads  GA  and  GB,  as  stated  above,  are  large 
in  cross-section  and  good  conductors,  and  hence 
show  no  energy  loss,  because  the  stress  per  molecular 
fiber  is  not  appreciable. 

The  1,000  watts,  or  1.34  horsepower,  which  is 
being  dissipated  in  heating  the  conductor  AB  is 


WORKING  HYPOTHESIS  23 

accounted  for  by  the  logical  assumption  of  work  done 
in  overcoming  resistance  to  the  rotation  of  molecular 
filaments. 

Current  direction  means,  in  this  case,  that  if  a  cross- 
section  of  the  conductor  be  taken  at  D,  the  direction 
of  rotation  of  the  molecular  filaments  is  clockwise 
when  viewed  in  the  direction  of  the  arrow. 

The  reference  to  the  "fall  of  potential"  around 
a  circuit  leads  us  to  inquire  for  an  analogous  phrase. 
Any  conclusion  drawn  in  this  respect  must  be  in 
harmony  with  the  statement  of  our  analogy  and 
with  our  working  hypothesis.  Study  of  the  former 
indicates  that  in  any  homogeneous  conductor  of 
uniform  size  the  stress  on  each  molecular  filament  is 
met  and  counterbalanced  by  resistance  to  rotation,  and 
this  resistance  is  uniformly  distributed  throughout  the 
length  of  the  filament. 

The  above  statement  may  be  applied  to  the  parts 
of  a  non-homogeneous  circuit,  as,  for  example,  one 
of  copper  wire  and  iron  wire  in  series. 

Greater  resistance  will  be  overcome  in  the  poorer 
conductor,  and  the  torque  indicator  (voltmeter)  will 
show  a  greater  reading  when  applied  to  the  iron 
part  of  the  circuit  than  when  applied  to  the  copper 
part. 

ELECTRIC  BATTERIES 

An  electric  generator  which  derives  its  electro- 
motive force  from  the  cutting  of  lines  of  magnetic 
flux  by  conductors  requires  a  prime  mover,  and  so 
becomes  an  uneconomical  machine  in  small  sizes. 


24  CHAPTER  III 

We  are  led,  therefore,  to  inquire  if  there  is  not  some 
other  means  of  instituting  molecular  torque.  Chem- 
ical action  is  immediately  suggested,  for  chemical 
energy  is  liberated  through  the  rearrangement  of 
molecules.  It  is  obvious  that  motion  must  accom- 
pany this  rearrangement  of  molecules,  and  an  as- 
sumption of  rotation  seems  not  illogical. 

This  prediction  is  justified  by  the  general  use  of 
elementary  chemical  generators  called  batteries. 

Commercial  batteries  are  of  two  general  classes, 
primary  and  secondary,  the  more  common  name  of 
the  latter  being  storage  batteries.  For  portable 
batteries,  the  dry  cell  has  been  developed.  The 
elements  of  which  it  is  composed  are  carbon,  zinc, 
and  a  solution  of  ammonium  chloride  made  into  a 
paste,  with  inert  ingredients  such  as  sawdust.  The 
"dry  cell"  is  merely  a  semi-wet  battery.  Wet  bat- 
teries are  used  where  portability  is  not  a  requisite, 
and  their  elements  may  be  replaced  when  chemical 
action  has  destroyed  any  of  them. 

In  most  primary  batteries  zinc  is  the  material 
which  is  actively  attacked  by  the  solution  in  which 
the  plates  are  immersed.  This  solution  is  called  the 
electrolyte,  and  in  some  batteries  more  than  one 
substance  is  used  in  the  electrolyte  to  counteract 
secondary  chemical  actions  which  hinder  the  main 
work  of  developing  voltage. 

Storage  or  secondary  batteries  are  those  in  which 
chemical  action  is  reversible.  Current  is  passed 
through  the  plates  and  solution  in  one  direction, 
causing  certain  changes  in  them  which  are  automati- 


WORKING   HYPOTHESIS  25 

cally  reversed  when  the  battery  is  used  to  supply 
current. 

One  type  of  storage  battery  has  lead  plates  in  an 
electrolyte  of  sulphuric  acid;  the  Edison  battery  has 
plates  of  nickel  hydroxide  and  iron  oxides,  and  uses  a 
solution  of  potassium  hydroxide  for  electrolyte.  The 
latter  type  of  battery  has  considerable  advantage 
over  the  lead  type  in  that  it  may  be  neglected  and 
abused  by  unskilled  persons  without  damaging  its 
capacity  for  absorbing  and  giving  off  energy.  The 
lead  type  of  cell  has  a  greater  immediate  reserve 
capacity  and  consequently  is  preferred  in  some  cases 
but  it  must  be  watched  carefully  on  charge  and 
discharge  to  prevent  serious  damage  to  the  plates. 

Storage  cells  are  rated  in  ampere  hours.  The 
ampere-hour  capacity  of  a  lead  cell  is  always  com- 
puted, unless  otherwise  specified,  on  the  basis  of  8- 
hour  discharge.  An  80  ampere-hour  cell  will  give 
10  amperes  for  8  hours,  and  10  amperes  is  the  normal 
charging  current.  Of  course,  a  greater  or  smaller 
current  can  be  taken  from  the  battery,  but  the 
ampere-hour  capacity  will  be  different.  For  ex- 
ample, if  a  higher  rate  be  chosen  and  20  amperes 
taken,  the  battery  will  only  last  about  3  hours; 
whereas  if  5  amperes  only  are  taken,  the  ampere- 
hour  capacity  will  apparently  be  increased,  and  it 
will  stand  up  for  perhaps  20  hours. 

This  makes  the  8-hour  rating  a  very  important 
point  to  know  in  buying  or  using  lead  storage  bat- 
teries. Edison  batteries  are  designed  and  rated 
ordinarily  on  a  7-hour  charge  and  5-hour  discharge. 


26  CHAPTER   III 

There  are  listed  below  the  voltages  of  some  common 
cells. 

Type  of  Cell  Voltage 

Dry  cell,  1.45 

Leclanche  wet  cell,  1 .45 

Lead  storage,  charged  2 . 5 

discharged  1 . 8 
Edison  storage,  charged  1 . 85 

discharged  1 . 00 

The  only  reliable  test  of  the  state  of  charge  of  a 
lead -acid  battery  is  the  density  reading  of  its  electro- 
lyte. On  full  charge  in  a  healthy  battery,  the  density 
should  be  1.25  to  1.30,  and  on  discharge  should  not 
fall  below  1.15  to  1.18.  Hydrometer  readings  must 
be  taken  carefully  with  respect  to  the  temperature 
of  the  electrolyte.  The  densities  here  given  cor- 
respond to  a  temperature  of  60°  Fahr.  A  gain  of 
0.001  is  caused  by  each  3°  drop  in  temperature,  and 
conversely,  a  loss  of  0.001  is  found  to  follow  an 
increase  of  3°  in  temperature. 


CHAPTER  IV 

Generation  of  Current  in  an  Armature  Coil — Current  alternat- 
ing as  generated.  Sine-Curve  Representation  of  Alternat- 
ing Current.  Changing  Alternating  Current  of  Armature  to 
Direct  Current  at  Machine  Terminals.  Principle  of  Com- 
mutator. Theory  of  Direct-Current  Motors. 


STUDENT'S  GUIDE 

The  manner  in  which  the  electric  generator  develops 
electromotive  force  is  deduced  in  this  chapter  by 
means  of  the  application  of  our  working  hypothesis. 
It  is  shown  that  the  induced  currents  of  the  arma- 
ture are  alternating,  and  the  analogy  is  employed  to 
reveal  the  use  and  operation  of  the  commutator  in 
rectifying  the  alternating  current  so  that  direct  cur- 
rent may  be  supplied.  A  brief  exposition  of  the  theory 
of  the  direct-current  motor  is  appended. 

How  PROCESS  SUGGESTED  BY  ANALOGY  Is 
INSTITUTED  AND  MAINTAINED 

It  is  well  known  that  when  a  closed  loop  of  copper 
wire  is  revolved  in  a  magnetic  field,  a  current  of 
electricity  is  set  up  and  maintained  in  the  wire.  When 
many  of  these  coils  are  wound  upon  a  spool  which 
revolves  in  a  magnetic  field  so  that  the  coils  of  wire 
cut  the  lines  of  force  in  the  proper  manner,  the  com- 
bination is  called  an  armature.  By  what  process  is  a 
current  generated  in  the  armature  coil  ?  Our  analogy 


28 


CHAPTER  IV 


calls  for  the  revolution  of  the  lines  of  molecules  in  a 
conductor  transmitting  energy.  It  also  suggests  the 
following  possible  way  in  which  such  revolution  may 
be  originated  and  maintained  in  the  conductor. 

In  Fig.  10,  let  N  and  5  be  the  north  and  south  poles 
respectively  of  a  field  magnet.  The  space  ABCD  is 
a  magnetic  field  of  force,  in  which  the  lines  of  force 


are  approximately  parallel  to  AB  and  DC.  Let  E  and 
F  represent  cross-sections  of  a  closed  coil  of  wire,  re- 
volvable  about  a  central  axis  perpendicular  to  the 
plane  of  section.  When  such  revolution  takes  place, 
lines  of  force  in  the  field  are  cut  by  the  coil,  and  a 
current  is  set  up  in  the  wire.  As  we  know  that  some 
process  goes  on  in  the  wire,  we  will  assume  as  a  work- 
ing hypothesis  that  while  the  wire  is  cutting  the  lines 


GENERATOR  29 

of  force  in  the  field  ABCD,  each  molecule  in  the  field 
is  magnetized  with  its  magnetic  axis  at  right  angles 
to  the  lines  of  force,  and  its  north  pole  in  the  general 
direction  of  the  motion  of  the  coil  at  the  point  where 
the  molecule  is  located. 

In  Fig.  10,  let  E  now  represent  a  particular  molecule 
in  one  side  or  branch  of  the  coil,  and  F  another  mole- 
cule in  the  opposite  side  of  the  coil.  E  and  F,  with 
their  subscripts,  represent  the  same  molecule,  but 
in  different  positions  of  the  coil.  As  E  is  carried  by 
the  revolution  of  the  coil  to  Ei,  we  shall  find  F  at 
FI.  By  our  hypothesis  the  molecule  in  the  position 
EI  is  now  being  magnetized  with  its  magnetic  poles  as 
shown  by  the  arrow  SN,  while  the  molecule  at  Fi  will 
have  its  poles  as  also  shown  by  its  arrow  SN.  In  each 
case  the  polar  axis  is  at  right  angles  to  the  lines  of 
force  of  the  magnetic  field,  and  the  north  pole  is  in  the 
general  direction  of  the  motion  of  the  wire  at  the 
point  where  the  molecule  is  situated.  Each  of  these 
tiny  magnets  EI  and  F\,  being  under  the  influence  of 
the  large  field  magnet  NS,  will  tend  to  revolve  rapidly 
and  in  such  a  direction  as  to  bring  the  north  pole  of 
the  molecule  toward  the  south  pole  of  the  field  magnet, 
and  vice  versa. 

Looking  at  the  figure  with  this  in  mind,  the  mole- 
cules Ei  and  FI,  which  are  supposed  to  be  both  in  and 
forming  a  part  of  the  same  flexible  shaft,  appear  to 
be  revolving  in  opposite  directions;  but  as  these  two 
molecules  are  in  opposite  branches  or  sides  of  the  same 
coil,  they  are  really  revolving  correctly  as  a  part  of  the 
same  flexible  revolving  shaft.  This  can  be  illustrated 


30  CHAPTER   IV 

by  bending  a  flexible  shaft  or  even  a  small  cord  into 
a  loop  STU  (Fig.  12)  and  twisting  the  cord  between 
the  thumb  and  finger  at  T.  Looking  from  T  toward 
EI  and  Fi  (Fig.  12),  the  cord  will  appear  to  revolve 
in  opposite  directions  at  the  points  E\  and  FI,  but  will 
really  be  revolving  throughout  its  length  harmoniously 
in  the  same  direction. 

When  the  coil  in  Fig.  10  has  revolved  through  180° 
from  its  first  position  EF,  so  that  the  molecule  E  has 
come  to  the  position  Ez  and  F  to  F%,  the  coil  ceases  to 
cut  lines  of  force  of  the  magnetic  field.  As  the  revolu- 
tion of  the  coil  proceeds,  it  begins  to  cut  again  the 
lines  of  force;  but  when  the  molecule  E  has  come  to  a 
position  EZ  and  F  to  Fs,  these  molecules  will  obviously 
be  each  revolving  in  a  direction  opposite  to  its  former 
motion.  The  rate  at  which  lines  of  force  are  cut  by 
the  revolving  coil  is  greatest  when  the  coil  is  90°  from 
the  position  EF.  This  rate,  starting  from  zero  at 
EF,  varies  as  the  sine  of  the  angle  through  which  the 
coil  has  revolved  from  EF.  This  appears  at  once 
from  the  inspection  of  the  figure.  Thus,  following 
our  analogy  in  connection  with  the  above  hypothesis, 
we  are  led  to  the  conclusion  that  the  electric  current 
is  alternating  in  the  armature,  and  that  its  direction 
changes  as  often  as  the  plane  of  the  coil  comes  into 
the  position  EF  or  E^;  that  is,  when  the  plane  of  the 
coil  is  perpendicular  to  the  lines  of  force  of  the  field. 

The  electromotive  force  thus  generated  may  con- 
veniently be  indicated  as  in  Fig.  11,  where  the  ordi- 
nates  of  the  curve  represent  the  instantaneous  values 
of  the  e.m.f .  and  the  abscissae  the  corresponding  angu- 


GENERATOR 


31 


lar  positions  of  the  rotating  coil.  This  is  the  so  called 
sine-wave,  the  properties  and  construction  of  which 
are  discussed  in  the  Appendix. 

It  should  be  noted  that  in  the  solution  of  the 
alternating-current  problems  the  sine-wave  is  taken 
to  represent  the  instantaneous  values  of  current  and 
voltage  because  calculations  are  thereby  made  simple, 
and  because  commercial  alternators  do,  in  fact,  give 
voltage  values  very  closely  approximating  the  theoreti- 


OF  ANGULAR 
POSITION,  OR  T/ME 


FIG.  11 

cal.  But  the  irregularities  in  the  distribution  of  the 
coils  about  the  armature,  and  variations  of  magnetic 
flux  distribution  in  the  air  gap,  sometimes  cause 
variations  from  the  true  sine-wave  form.  And  when 
a  pure  sine-wave  voltage  is  impressed  upon  an  induct- 
ance coil  having  an  iron  core,  the  current  has  a  wave 
form  considerably  distorted  from  the  sine-curve. 

In  the  Appendix  will  be  found  definitions  of  form 
factor  and  peak  factor.  These  quantities  are  measures 
of  the  variation  of  the  wave  from  the  sine  form. 


32 


CHAPTER  IV 
THE  COMMUTATOR 


Having  shown  that  the  current  as  generated  in  the 
dynamo  is  always  alternating,  we  refer  to  our  analogy 
to  show  how  a  direct  current  is  obtained  in  the  external 


U 


circuit,  from  the  alternating  current  generated  in  the 
armature  of  the  dynamo. 

In  Fig.  13,  let  ABCE  be  an  endless  flexible  shaft. 
If  this  shaft  is  given  a  rotary  motion  at  any  point  as 
A,  the  shaft  will  rotate  through  its  whole  length.  This 
illustrates  a  direct  current  of  electricity.  If  the  rotary 


GENERATOR 


33 


motion  induced  at  A  be  first  in  one  direction  and  then 
in  the  opposite,  this  alternating  motion  at  A  will 
obtain  throughout  the  whole  shaft,  and  we  shall  have 
illustrated  an  alternating  current  in  ABCE.  Now 

C 


suppose  that  at  E  and  B  there  are  detachable  con- 
nections. 

In  Fig.  13,  let  the  rotation  at  A  be  in  a  given  direc- 
tion. Then  the  rotation  of  BCE  will  be  in  the  same 
direction  as  the  rotation  at  A.  Now  let  the  connec- 
tions at  E  and  B  be  detached  and  E  and  B  change 
places  with  reference  to  the  ends  of  the  part  C,  as 


34  CHAPTER   IV 

shown  in  Fig.  14;  and  at  the  same  instant  let  the  direc- 
tion of  the  rotation  at  A  be  changed.  Then  it  readily 
appears  that  while  the  rotation  at  A  alternates,  the 
rotation  of  BCE  will  continue  in  the  same  direction 
as  before.  This  interchange  back  and  forth  between 
E  and  B,  occurring  as  often  as  and  simultaneously 
with  the  change  in  direction  of  rotation  at  A,  will  give 
a  continuous  rotary  motion  in  the  coil  C;  or  in  other 
words,  the  alternating  current  at  A  has  been  changed 
into  a  direct  current  in  the  external  circuit  C.  This 
reveals  at  once  the  principle  of  the  commutator. 

The  continuous  repetition  of  changes  back  and 
forth  from  EB  to  BE  is  secured  in  the  direct-current 
dynamo  by  the  proper  connections  of  the  armature 
wires  with  the  commutator  sections.  The  direct 
current  is  taken  from  the  commutator  by  the  brushes, 
which  are  connected  with  the  external  circuit  con- 
ductors. 

The  principle  of  the  commutator  enables  one  to 
understand  the  rotary  converter,  which  is  sub- 
stantially an  armature  receiving  by  means  of  sliding 
contacts  alternating  current  from  any  source  and 
changing  it  into  direct  current,  or  vice  versa,  in  the 
same  manner  as  the  commutator  of  the  dynamo 
changes  the  alternating  current  received  from  the 
armature  wires  into  the  direct  current  of  the  external 
circuit. 

THEORY  OF  DIRECT-CURRENT  MOTORS 

If  an  electric  current  is  established  in  a  conductor 
lying  in  a  magnetic  field,  mechanical  force  will  be 


GENERATOR 

exerted  on  the  conductor,  the  strength  of  which  will  be 
proportional  to  the  strength  of  the  field  and  the 
strength  of  the  current.  It  will  be  seen  that  the  direct- 
current  generator  can  thus  be  reversed  and  used  as  a 
motor  simply  by  applying  to  its  terminals  an  electro- 
motive force  obtained  from  some  other  similar  machine 
which  is  being  driven  by  a  steam  or  hydraulic  prime 
mover. 

Figure  15  shows  the  circuits  of  a  common  type 
known  as  the  shunt  motor.    The  source  of  electric 


0 

FIG.  15 

potential  is  shown  at  DD.  The  magnetic  excitation 
of  the  field  poles  of  the  motor  is  furnished  by  the  coil 
F,  with  its  strength  regulated  by  a  variable  resistance 
R.  The  main  current  flows  into  the  rotating  armature 
A  through  brushes  B. 

Let  us  examine  one  of  the  conductors  on  the  arma- 
ture in  order  that  we  may  discover  the  manner  in 
which  the  armature  current  is  adjusted  to  meet  load 
requirements.  The  armature  A  and  the  brushes  B 
are  of  low  resistance;  and  if  there  were  no  other  opposi- 
tion offered,  the  current  through  the  armature  would 
be  unlimited  and  would  bear  no  relation  to  the  load ; 
that  is,  to  the  torque  required  at  the  motor  pulley. 


36  CHAPTER   IV 

Figure  16  is  a  section  of  the  armature  and  a  pole- 
face,  with  the  magnetic  field  represented  by  lines  in 
the  air  gap.  A  single  conductor  P  is  shown.  If  the 
direction  of  rotation  of  filaments  is  left  handed,  with 
the  magnetic  polarity  shown,  application  of  the  work- 
ing hypothesis  (left-hand  rule,  see  Chapter  V)  gives 
the  direction  of  rotation  of  the  armature  shown  by 
the  arrow. 

Now  conductor  P  is  cutting  magnetic  lines  and 
therefore  must  have  induced  in  it  an  electromotive 


FIG.  16 

force.  The  working  hypothesis  (right-hand  rule) 
gives  the  direction  of  this  induced  torque  as  right- 
hand  rotation,  and  so  we  see  that  the  induced  voltage 
is  tending  to  oppose  the  voltage  which  is  sending  the 
motor  current  through  P  in  the  left-hand  direction. 
Suppose  that  a  load  is  suddenly  removed  from  the 
motor;  there  will  then  be  an  excess  of  mechanical 
torque,  and  the  armature  speed  will  increase  until  the 
counter  e.m.f.  has  reduced  the  current  to  the  value 
just  large  enough  to  supply  the  new  reduced  me- 


GENERATOR  37 

chanical  torque.  When  the  load  is  applied  again,  the 
armature  speed  drops,  the  back  e.m.f.  is  thereby 
reduced,  and  more  armature  current  flows  to  provide 
the  needed  mechanical  torque.  In  this  way  the  back 
e.m.f.  is  always  opposing  the  applied  voltage,  acting 
as  an  automatic  resistance,  adjusting  itself  to  load 
requirements. 

Current  taken  by  the  armature  is  represented  by 
the  following  expression 


where 

I  a    =  armature  current; 

EL  =  voltage  at  brushes; 

Ea  =  counter  e.m.f.  of  armature; 

Ra  =  resistance  of  armature. 


CHAPTER  V 

Induction — Relation  of  magnetism  and  electricity;  relation  of 
induced  to  primary  voltage;  hand  rules  for  determining 
direction  of  induced  effects.  Self-induction — Example; 
response  of  molecular  filaments  to  changes  in  strength 
of  magnetic  field.  Transformer — examples  of  use;  voltage, 
current,  and  energy  relations.  Induction  Coil — Alternating 
current  induced  by  pulsating  direct  current. 


STUDENT'S  GUIDE 

Self  and  mutual  induction  are  the  main  headings 
for  this  chapter.  We  here  expand  our  knowledge  of 
the  function  of  magnetism  in  electrical  phenomena, 
and  learn  that  the  molecular  filaments  of  a  conductor 
are  at  all  times  subject  to  very  delicate  control  by 
magnetic  lines  of  force.  Illustrations  chosen  are  the 
telephone,  the  transformer,  and  the  induction  coil. 

INDUCTION 

When  a  conductor  in  circuit  and  a  magnetic  field  of 
force  are  in  such  mutual  relations  that  the  conductor 
moves  relatively  to  the  lines  of  force  in  such  a  manner 
as  to  cut  these  lines,  a  current  of  electricity  is  gen- 
erated in  the  conductor.  This  action  is  called  induc- 
tion. The  lines  of  force  may  be  constant  and  the 
conductor  move  across  them,  or  the  conductor  may 
be  at  rest  but  in  a  fluctuating  field  of  force. 

In  all  cases  the  statement  of  our  working  hypothe- 
sis holds — namely,  when  a  conductor  in  a  magnetic 


INDUCTION 


39 


field  of  force  cuts  these  lines  of  force,  the  molecules 
of  the  conductor  constantly  tend  to  form  magnetic 
poles,  their  instantaneous  polar  axes  being  at  right 
angles  to  the  lines  of  force  of  the  magnetic  field,  and 
with  the  north  pole  in  the  general  direction  of  the 
motion  of  the  conductor  relatively  to  the  magnetic 
lines  of  force.  By  means  of  this  hypothesis  we  may 
readily  explain  the  phenomena  of  induction. 

Referring  to  Chapter  I,  we  find  that  a  conductor 
carrying  a  current  seems  to  be  surrounded  by  lines 


FIG.  17 

of  magnetic  force;  a  magnetic  needle  placed  near  it 
always  tends  to  stand  at  right  angles  to  the  conductor. 
If  the  current  changes  direction,  the  magnetic  needle 
instantly  tends  to  change  its  direction  through  180°, 
If  our  assumption  is  correct  that  the  molecules  of  the 
conductor  are  constantly  magnetized  as  stated,  then 
the  effect  upon  these  molecules  in  the  presence  of  a 
conductor  carrying  a  current  will  be  the  same  as  the 
effect  upon  a  magnetic  needle. 


40  CHAPTER   V 

In  Fig.  17  let  A  be  a  cross-section  of  a  conductor 
carrying  an  alternating  current,  and  let  B  be  a  con- 
ductor in  another  circuit,  the  two  conductors  A  and 
B  being  parallel  and  near  each  other.  Let  the  con- 
centric circles  about  A  represent  the  magnetic  lines 
of  force  which  we  know  to  exist  around  a  conductor. 
Since  the  current  in  A  is  alternating,  these  lines  of 
force  will  go  from  zero  to  their  maximum  distance 
from  A  and  back  to  zero,  as  often  as  the  current  alter- 
nates. In  consequence  of  this,  a  current  will  be  set 
up  in  conductor  B,  usually  called  an  induced  current. 


FIG.  18 

What  follows  explains  how  the  e.m.f.  of  this  induced 
current  may  be  generated,  and  why  it  is  90°  behind 
the  electromotive  force  of  the  conductor  A . 

Let  Fig.  18  represent  the  electromotive  forces  of 
the  current  in  the  conductor  A  and  of  the  induced 
current  in  the  conductor  .B,  in  Fig.  17,  and  explained 
above.  The  letters  L,  M,  N,  0,  P,  Q,  in  Fig.  18  repre- 
sent periods  in  the  cycle  90°  apart,  with  L  beginning 
at  zero.  Let  Ci,  £2,  G,  represent  the  electromotive 
force  of  the  current  in  A .  As  the  electromotive  force 
goes  from  the  point  C\  at  the  time  M  (90°  from  the 
beginning  of  the  cycle)  to  zero  degrees  at  time  N 


INDUCTION  41 

(180°  from  the  beginning  of  the  cycle),  the  lines  of 
force  from  the  conductor  A  of  Fig.  17  cut  across  the 
conductor  B  of  Fig.  17  as  they  would  if  B  moved 
towards  the  circumference  of  these  lines  relatively  to 
the  lines;  and  consequently  the  molecules  of  the  con- 
ductor B  are  magnetized  with  their  polar  instanta- 
neous axes  at  right  angles  to  the  lines  of  force,  and  the 
north  poles  of  these  molecules  pointing  away  from  A . 
The  molecules  of  B  therefore  tend  constantly  to  rotate 
into  a  position  with  their  polar  axes  at  right  angles 
to  the  conductor  A.  Consequently  a  current  will  be 
set  up  in  the  conductor  B,  whose  electromotive  force 
is  represented  by  the  line  Dit  D2,  £>3,  in  Fig.  18. 

At  the  point  marked  N  in  Fig.  18  (180°  from  the 
beginning  of  the  cycle)  the  current  in  A  alternates, 
and  also  the  motion  of  the  conductor  B  relatively  to 
the  lines  of  force  around  A,  Fig.  17,  changes  direction. 
The  alternation  of  the  current  changes  the  polarity  of 
the  conductor  .4,  and  the  polar  axes  of  the  molecules  of 
the  conductor  B  also  change  through  180°  at  the  same 
instant.  And  similarly  at  the  point  P  in  Fig.  18. 
Therefore  the  electromotive  force  in  conductor  B  in 
Fig.  17  will  be  represented  in  Fig.  18  by  the  line  Di, 
D2)  DZ\  or  in  other  words,  the  molecules  in  the  con- 
ductor B  will  continue  to  rotate  in  the  same  direction 
from  M  to  0,  or  during  the  period  from  90°  to  270° 
of  the  cycle;  and  similarly,  while  the  electromotive 
force  of  conductor  A  passes  from  0  to  Q,  the  molecules 
in  conductor  B  will  rotate  in  the  opposite  direction. 
Hence  the  electromotive  force  of  the  induced  current 
is  90°  behind  that  of  the  original  current. 


42  CHAPTER   V 

It  will  be  well  at  this  point  to  state  two  rules  which 
have  been  found  to  apply  to  the  phenomena  of  in- 
duction. It  is  highly  important  that  we  be  able  to 
predetermine  the  direction  of  induced  currents.  Ordi- 
narily the  student  of  electricity  is  cautioned  to  think 
only  of  induced  electromotive  force  or  voltage,  current 
itself  never  being  said  to  be  induced,  but  always  con- 
sidered as  resulting  from  the  action  of  voltage  on  a 
resistance.  But  the  definition  of  electricity  as  a  means 
of  transmitting  energy,  and  the  conception  of  the 
generation  of  current  by  the  forcible  rotation  of  ma- 
terial molecular  filaments,  make  the  distinction  less 
important  and  consequently  we  shall  use  the  phrases 
"induced  voltage"  and  " induced  current"  as  may 
best  suit  our  convenience. 

There  are  two  "hand  rules"  which  are  helpful  in 
applying  our  working  hypothesis  when  determining 
the  direction  of  induced  current  in  generator  action, 
and  the  direction  of  motion  when  motor  action  is 
taking  place.  The  "right-hand  rule"  applies  to  the 
generator,  or  to  any  case  of  induced  current  where  it 
is  possible  to  determine  the  relative  motion  of  con- 
ductor and  lines  of  force. 

Right-Hand  Rule— Place  the  right  hand  in  the 
magnetic  field  with  thumb  and  fingers  at  right  angles 
and  extended,  so  that  the  lines  of  force  enter  the  palm 
of  the  hand,  with  the  thumb  indicating  the  direction 
of  relative  motion  of  the  conductor;  the  fingers  will 
then  indicate  the  direction  of  the  induced  current. 

For  motor  action  the  "left-hand  rule"  applies. 

Left-Hand  Rule — Place  the  left  hand  in  the  mag- 


INDUCTION  43 

netic  field  with  thumb  and  fingers  at  right  angles  and 
extended,  so  that  the  lines  of  force  enter  the  palm  of 
the  hand,  with  the  fingers  indicating  the  direction  of 
the  current;  the  thumb  will  then  indicate  the  direction 
of  the  mechanical  force  exerted  on  the  conductor. 


SELF-INDUCTION 

In  the  preceding  discussion  the  assumption  has 
been  made  that  two  circuits  were  always  concerned, 
one  having  a  current  flowing  in  it  producing  a  mag- 
netic flux  which,  linking  with  the  second  circuit, 
induced  therein  a  current.  The  phenomenon  of 
induction  is  also  observed  in  a  single  circuit  whenever 
the  current  strength  changes,  for  with  every  change 
of  current  a  change  of  magnetic  field  strength  occurs, 
the  circuit  is  cut  by  its  own  lines  of  magnetic  force,  and 
is  subject  to  the  several  laws  of  induction. 

The  practical  effect  of  this  action  may  be  observed 
on  breaking  the  current  in  a  highly  self-inductive 
circuit — that  is,  a  circuit  in  which  the  mechanical 
arrangement  is  such  as  to  allow  many  magnetic  lines 
to  cut  the  conductors  of  the  circuit.  The  change 
which  is  being  effected  is  a  decrease  of  current,  and 
the  torque  with  which  self-induction  opposes  this 
change  causes  the  arcing  at  the  switch  points  which 
is  characteristic  of  inductive  circuits,  such  as  the  field 
windings  of  generators.  If  we  were  dealing  with  the 
rotation  of  flexible  shafting  of  finite  size,  the  tendency 
to  continue  in  motion  would  without  hesitation  be 
ascribed  to  inertia;  and  we  are  thus  inclined  to  endow 


44  CHAPTER  V 

our  molecular  filaments  with  a  property  of  the  nature 
of  inertia.  But  we  must  recognize  the  fact  that  it  is 
not  a  phenomenon  comparable  with  that  of  a  body  of 
sensible  dimensions,  for  it  is  a  fact  of  common  experi- 
ence that  the  voltage  which  appears  at  the  terminals 
of  a  field  circuit  when  the  field  current  is  suddenly 
interrupted  is  often  many  times  greater  than  the 
voltage  impressed  on  the  field  when  the  normal  field 
current  is  flowing  without  interruption. 

The  energy  thus  exhibited  is  stored  in  the  magnetic 
field  while  the  current  is  flowing  steadily  in  the  coils 
and  is  not  due  to  any  tendency  of  the  filaments  to 
continue  their  motion,  but  to  a  true  electromotive 
force  or  torque  which  is  generated  in  the  manner 
described  above,  because  of  the  collapsing  of  the  lines 
of  force  upon  the  turns  of  the  field  coils,  after  the 
switch  is  opened. 

It  is  fitting  at  this  time  to  point  out  the  fluidity  of 
motion  which  the  filaments  of  a  good  conductor 
possess.  They  are  instantly  and  definitely  responsive 
to  any  torque  impressed  by  the  slightest  variation  of 
the  magnetic  field.  This  relation  is  shown  very  clearly 
in  the  operation  of  the  telephone  transmitter,  which 
consists  of  a  soft  iron  diaphragm  mounted  near  the 
poles  of  a  horseshoe-type  permanent  magnet  which 
is  provided  with  coils  of  wire  wound  on  the  poles.  The 
attraction  of  the  magnet  holds  the  diaphragm  in 
slight  tension.  When  a  sound  disturbs  the  diaphragm 
and  causes  it  to  vibrate,  the  magnetic  field  strength 
is  varied  as  the  diaphragm  approaches  or  recedes 
from  the  magnet  poles.  This  variation  of  field  causes 


INDUCTION  45 

a  cutting  of  the  turns  of  wire  in  the  coils  by  lines  of 
force,  and  currents  are  caused  to  flow  in  the  coils  and 
in  the  telephone  line.  The  receiver  at  the  other  end 
of  the  line  is  constructed  similarly,  and  the  reverse 
process  is  instituted,  causing  the  diaphragm  to 
vibrate  in  unison  with  that  of  the  transmitter  and  to 
reproduce  the  sound  which  caused  the  original 
vibrations.  The  transmitter  as  used  to-day  is  not 
of  the  type  described  above,  but  the  receiver  has  not 
been  modified  except  in  form  and  arrangement  of  the 
parts  described. 

The  foregoing  discussion  suggests  the  utilization 
of  the  principle  of  induction  in  an  apparatus  which, 
by  an  increase  or  decrease  of  the  number  of  conductors 
in  the  circuit  in  which  the  induced  current  flows,  can 
raise  or  lower  the  voltage  as  desired,  with  a  conse- 
quent change  of  the  angular  velocity  (or  current)  to 
keep  the  energy  equation  balanced.  The  effect  of  the 
transformer  is  like  that  of  the  spur  and  pinion  of 
mechanical  transmission  systems. 

If  in  Fig.  17  we  place  another  conductor  C  beside 
B,  as  in  Fig.  19,  there  will  be  induced  in  it  an  electro- 
motive force  exactly  equal  and  in  phase  with  that 
induced  in  B.  Should  these  two  conductors  be  con- 
nected in  series,  we  should  have  a  two  to  one,  step-up 
transformer  action.  The  current  in  BC  will  be 
one-half  that  in  A,  in  order  that  the  energy  relation 
may  be  a  true  one. 

The  distribution  of  energy  in  lighting  a  city  is 
accomplished  by  means  of  transformers.  In  a  specific 
case  the  hydroelectric  generating  plant  is  200  miles 


46 


CHAPTER  V 


from  the  cities  where  the  energy  is  used.  The  genera- 
tors operate  at  2,300  volts,  and  their  output  is  com- 
municated to  the  transmission  line  by  means  of 
transformers  which  change  the  voltage  to  100,000, 
with,  of  course,  a  proportionate  drop  in  current. 

It  is  the  lowering  of  current  which  is  desired,  for 
the  line  losses  due  to  heating  are  proportional  to  the 
square  of  the  current. 

On  the  outskirts  of  the  cities  there  are  transforming 
stations  where  the  reverse  is  accomplished,  the  trans- 


FIG.  19 

formation  being  from  100,000  down  to  2,200  volts, 
for  it  would  be  dangerous  to  carry  the  higher  voltage 
through  the  streets  and  into  buildings.  Each  con- 
sumer of  energy  then  has  his  own  smaller  transformer, 
which  further  reduces  the  voltage  to  220  for  motors 
and  110  for  house  lighting. 

The  facts  to  be  remembered  about  transformers 
are,  that  the  ratio  of  transformation  of  voltage  is 
directly  as  the  number  of  turns  in  primary  and 
secondary  coils;  that  if  voltage  is  raised,  current  is 


INDUCTION  47 

lowered ;  and  that  some  energy  is  always  lost  through 
heating  the  conductors  and  the  iron  core  on  which 
they  are  wound. 

These  relations  are  expressed  thus: 

Ni  =  number  of  turns  in  primary  winding ; 

Nz  =  number  of  turns  in  secondary  winding; 

/i     =  current  in  primary; 

/2     =  current  in  secondary; 

Ei    =  voltage  impressed  on  primary; 

EZ    =  voltage  induced  in  secondary; 

Wi  =  energy  input; 

W8  =  energy  lost  in  heating  transformer; 

Wz  =  energy  taken  from  secondary. 

Nl        El  *T    , 

-  --  EJ, 

(assuming  non-inductive  load  and  neglecting  losses). 
Wi  =  W.  +  W* 

Wz 
Efficiency  =  — - 

Voltage  and  current  relations  in  a  transformer  are 
related  by  the  laws  stated  at  the  beginning  of  this 
chapter.  (See  Chapter  VI  for  transformer  connec- 
tions.) 

The  induction  coil  is  a  common  and  useful  piece 
of  apparatus  in  which  a  pulsating  direct  current  of 
low  voltage  is  converted  to  an  alternating  current  of 
high  voltage.  Figure  20  shows  the  circuits  of  a  simple 
induction  coil. 

C  is  a  soft  iron  core  on  which  are  wound  perhaps 
one  hundred  turns  of  medium-size  wire  P,  called  the 


48 


CHAPTER   V 


primary  winding.  One  end  of  the  primary  is  brought 
out  through  /,  a  vibrating  armature  type  of  interrup- 
ter which  causes  rapid  interruption  of  the  primary 
current.  S  is  the  secondary  winding  which  consists 
of  many  hundreds  of  turns  of  fine  wire  wound  over 
the  primary  turns. 

When  the  current  is  first  established  in  the  primary, 
lines  of  force  spread  from  the  turns  of  the  primary 
coil  and,  in  doing  so,  cut  the  turns  of  the  secondary, 
thus  inducing  a  high  voltage  which  appears  at  the 
terminals  of  the  secondary  winding.  The  armature 


1 

c 

0 

• 

Is 

T"*      ^^ 

c 

> 

T 

0 

e 

,  -^ 

i 

• 

9 

FIG.  20 


is  attracted  by  the  core  C,  thus  interrupting  the 
current  and  causing  the  collapse  of  the  lines  of  force. 
In  collapsing  they  cut  the  secondary  turns  in  the 
opposite  direction,  and  the  electromotive  force  at  the 
terminals  is  equal  to  that  induced  when  the  current 
was  established,  but  opposite  in  direction.  Conse- 
quently for  each  to  and  fro  motion  of  the  armature  or 
interrupter,  a  complete  cycle  of  alternating  electro- 
motive force  is  induced  in  the  secondary  winding. 
The  relation  between  current  in  the  primary  C  and 
the  induced  secondary  voltage  E  is  shown  in  Fig.  21. 
In  order  that  the  student  may  have  some  familiarity 


INDUCTION 


49 


with  the  dimensions  and  materials  necessary  for  con- 
structing the  above  described  coil,  there  is  given  here 


FIG.  21 

a  list  of  the  parts  of  a  coil  which  will  produce  at  its 
secondary  terminals  a  spark  3  inches  long.  Core: 
bundle  of  iron  wire,  annealed  No.  20,  1J4  inches  in 


50  CHAPTER   V 

diameter,  13  inches  long;  primary,  four  layers  of  No. 
12  double  silk  covered  copper  wire,  about  4J/2  pounds; 
secondary,  4  pounds  of  No.  36  double  silk  covered 
copper  wire,  about  28,000  feet.  This  coil  will  have  an 
effective  voltage  of  approximately  100,000  volts  at 
the  secondary  terminals. 


CHAPTER  VI 

Combining  Out  of  Phase  Torques — Graphical  solution;  triangle 
relation  between  Z,  R,  and  S.  Power  Factor — Adjustment 
of  power  factor  by  introduction  of  new  torque;  efficiency 
of  this  procedure.  Three-Phase  Circuits  Developed — 
Three-phase  alternator;  transformers  connected  to  three- 
phase  alternator.  Measurement  of  three-phase  power — 
Two- wattmeter  method. 


STUDENT'S  GUIDE 

We  have  seen  that  several  torques  out  of  phase  with 
each  other  may  be  acting  on  a  circuit  at  once;  in  this 
chapter  we  shall  show  how  to  combine  them.  The 
relation  between  certain  of  these  original  and  resultant 
torques  is  important  enough  to  be  named — we  call 
this  relation  the  power  factor;  and  since  it  is  desired 
to  have  the  highest  obtainable  power  factor,  we  out- 
line means  of  establishing  the  desired  relation.  From 
the  general  case  of  out  of  phase  torques  it  is  but  a  step 
to  the  orderly  arrangement  of  three-phase  systems, 
and  the  chapter  closes  with  a  discussion  of  the  means 
of  measuring  the  power  in  three-phase  transmission 
circuits. 

COMBINING  OUT  OF  PHASE  TORQUES 

An  electric  circuit  may  have  both  resistance  and 
self-induction.  Let  us  examine,  from  the  viewpoint 
of  our  analogy,  the  characteristics  of  a  circuit  having 
both  resistance  and  self-induction.  In  Fig.  22,  let 


52 


CHAPTER   VI 


the  sine-curve  L  represent  the  e.m.f.  or  torque  of  the 
resistance  or  ohmic  load.  Let  /  represent  the  e.m.f. 
of  self-induction,  and  let  the  current  be  a  unit  current. 
This  unit  current  should  have  a  maximum  ordinate 
which  gives  the  mean  value  equal  to  unity.  The 
curves  marked  C  represent  unit  currents.  It  has  been 


FIG.  22 

shown  that  an  induced  e.m.f.  is  90°  behind  the  prim- 
ary e.m.f.  Therefore  the  curve  /  starts  at  90°  while 
the  curve  L  starts  at  (zero  degrees)  0°.  Consider  a 
line  of  shafting  of  negligible  weight  and  acted  upon  by 
two  torques  that  are  out  of  phase  with  each  other. 
If  there  is  alternating  reciprocating  rotary  motion 
of  the  shaft  under  the  action  of  an  impressed  torque, 


OUT   OF   PHASE   TORQUES  53 

it  follows  that  the  impressed  torque  must  be  at  every 
instant  equal  and  opposite  to  the  resultant  of  the  two 
resisting  torques,  because  action  and  reaction  must 
be  equal  and  opposite.  It  also  follows  that  the  angular 
velocity  of  the  shaft  in  question  must  be  in  phase  with 
the  resultant  torque. 

Combine  graphically  the  torques  represented  by  L 
and  /.  The  resulting  curve  M  cuts  the  time  line  about 
18.27°  behind  the  point  at  which  L  cuts  it.  In  other 
words,  the  resultant  torque  has  been  set  back  this 
number  of  degrees  by  the  introduction  of  the  out  of 


phase  torque  of  self-induction.  The  sine-curves 
L,  M,  and  /  are  really  energy  curves,  since  the  current 
is  unity.  Their  areas  represent  respectively  the  im- 
pressed energy,  the  energy  required  to  overcome  the 
load,  and  the  energy  required  to  overcome  self-induc- 
tion, for  half  a  cycle,  and  are  proportional  to  the 
squares  of  the  maximum  ordinates  of  the  curves. 
(See  Appendix.)  Denoting  the  maximum  ordinates 
of  L,  /,  and  M  by  R,  S,  and  Z  respectively,  we  have 
Z2  =  R2  +  S2,  or  Z  =  \/R2  +  S2.  This  equation  is 
represented  graphically  by  the  triangle,  Fig.  23. 

In   this  triangle   the  angle  between   Z  and  R  is 
usually  termed  the  "lag  angle."    Let  it  be  denoted  by 


54  CHAPTER   VI 

6.  If  we  determine  this  angle  by  putting  the  values 

$ 

of  R  and  5  from  Fig.  22  into  the  equation  tan  6  =  — , 

R 

we  find  that  the  value  thus  computed  agrees  sub- 
stantially with  the  18.27°  shown  in  Fig.  22. 

The  meaning  of  the  term  "lag"  evidently  is  that 
the  current  lags  this  number  of  degrees  behind  the 
e.m.f.  of  one  of  the  component  torques — namely,  L — 
and  not  that  it  lags  behind  the  resultant  e.m.f.,  M. 

The  cosine  of  the  so  called  lag  angle  is  used  as  the 
power  factor,  and  its  use  for  this  purpose  is  very  con- 
venient. The  power  factor  is  the  relation  of  the  useful 
work  to  the  total  energy  expended.  Calling  R  or  the 
ohmic  load  the  useful  work,  which  is  represented  by 
the  area  of  L,  and  Z  the  energy  expended  as  repre- 
sented by  the  area  of  M,  we  should  have  power 

TO 

factor   =   — ,  which  gives  the  result  obtained  for  0 
Zi 

from  the  triangle,  Fig.  23. 

That  any  change  in  the  load  will  affect  6  is  evident 
from  the  accompanying  figures. 

Figure  24  shows  the  addition  of  non-inductive  load, 
with  consequent  decrease  of  6.  Figure  25  shows  the 
diagram  for  a  circuit  having  capacity  and  non-induc- 
tive resistance.  Figure  26  shows  a  case  of  inductive 
effect  overbalanced  by  capacity. 

An  important  situation,  which  is  discussed  in  the 
following  pages,  is  illustrated  by  Fig.  27.  Here,  by 
the  expenditure  of  a  little  additional  energy,  we  are 
able  to  introduce  into  the  circuit  a  torque  exactly 


OUT   OF   PHASE   TORQUES 


55 


equal  and  opposite  to  the  torque  of  self-induction, 
thus  bringing  0  to  zero,  the  power  factor  to  unity,  and 
increasing  the  output  of  the  system. 


^ 


FIG.  27 

The  angle    6   in    the  above    figures  —  commonly 
termed  the  "lag"  angle — is  due  to  the  self-induction 


56 


CHAPTER  VI 


being  out  of  phase  with  the  load.  It  is  an  indication 
rather  than  a  cause  of  the  loss  due  to  self-induction. 
A  way  to  bring  the  power  factor  to  unity  at  once 
suggests  itself  from  the  foregoing  analysis.  Let  Fig. 
28  represent  an  alternating-current  motor  the  arma- 
ture coils  of  which  form  a  part  of  the  circuit.  Let  the 
field  magnets  of  the  motor  be  energized  by  a  direct 
current  generated  by  a  dynamo  receiving  power  either 
from  an  outside  source  or  from  the  line  itself.  If 


FIG.  28 

when  the  coil  is  in  the  position  A — B,  its  current  and 
e.m.f.  are  at  a  maximum,  with  the  shafts  at  A  having 
counter-clockwise  rotation,  then  when  the  coil  is  in 
the  position  C — D,  or  90°  from  A — B,  the  e.m.f.  in- 
duced by  passing  the  coil  through  the  magnetic  field 
between  N  and  5  will  be  at  its  maximum,  the  shafts 
at  D  having  clockwise  rotation;  and  we  shall  have 
introduced  into  the  line  a  new  torque  which  may  be  at 
every  instant  equal  in  value  and  opposite  in  direction 


OUT  OF  PHASE  TORQUES  57 

to  7,  Fig.  22.  The  adjustment  required  to  thus  bring 
the  line  into  synchronism  consists  in  regulating  the 
strength  of  the  poles  N,  S  by  varying  the  exciting 
current.  This  is  a  crude  illustration  of  a  means  for 
securing  synchronism,  but  it  makes  clear  the  principle 
of  the  synchronous  condenser  on  the  line. 

With  such  an  arrangement  as  the  above,  the  line 
is  sometimes  said  to  have  capacity.  The  effect  is 
such  as  would  result  from  capacity,  but  the  process  is 
not  in  accord  with  our  idea  of  capacity.  (See  Chapter 
VII.)  The  result  of  the  condenser  on  the  line  as  above 
described  is  shown  in  Fig.  22  by  the  curve  /i,  which 
is  equal  to  the  curve  I,  but  180°  behind  /.  Con- 
structing now  the  curve  representing  the  resultant 
e.m.f.,  we  find  it  to  be  the  curve  L.  The  two  e.m.f.'s 
180°  apart  have  balanced  each  other  and  eliminated 
the  lag.  This,  however,  has  been  done  by  an  ex- 
penditure of  energy.  If  energy  is  taken  from  the  line 
to  run  a  motor  generator  set  which  supplies  direct 
current  to  energize  the  magnets  N  and  S  in  Fig.  28, 
this  will  add  to  the  load  L.  This  additional  load  may 
be  represented  by  a  sine-curve  /,  Fig.  22,  the  area  of 
which  should  be  somewhat  greater  than  the  area 
M — L.  To  illustrate  the  effect  of  this  additional  load, 
assume  that  it  is  once  and  a  half  M — L,  or 

/  =  —  (M — L.)    If  now  we  proceed  to  construct  the 

resultant  e.m.f.,  we  have  the  curve  L\.  The  line  is  now 
in  synchronism  and  carrying  a  load  and  a  resultant 
expenditure  of  energy  greater  than  the  original  load 
by  an  amount  equal  to  LI — L  and  greater  than  M  by 


58  CHAPTER  VI 

the  amount  L\ — M.     The  value  of  — —     rep  re - 

Lt\  — L* 

sents  the  efficiency  of  the  motor  generator  set  used 
to  energize  the  field  magnets  of  the  condenser  on  the 
line. 

Thus  by  expending  an  amount  of  energy  equal  to 
LI — L  we  have  produced  isochronism,  thus  overcom- 
ing the  effect  of  self-induction  and  raising  the  energy 
in  the  line  from  L  to  L\  and  increasing  the  net  output 
from  L  to  M\  =  M. 

THE  FOREGOING  RESULTS  ARE  LOGICAL 
DEDUCTIONS  FROM  OUR  ANALOGY  AND 
WORKING  HYPOTHESIS.  THESE  RESULTS 
ARE  NOT,  WE  BELIEVE,  ESSENTIALLY  CON- 
TRARY TO  OR  INCONSISTENT  WITH  THE 
LATEST  KNOWLEDGE  OR  THE  BEST  PRAC- 
TICE OF  ELECTRICAL  ENGINEERING.  THIS 
CHAPTER  IS  BUT  ONE  ILLUSTRATION  OF 
THE  PURELY  DEDUCTIVE  METHOD  WE 
ARE  ABLE  TO  EMPLOY.  IT  IS  WORTHY  OF 
NOTE  THAT  IN  OUR  DEDUCTIONS  WE  WERE 
NOT  DEPENDENT  UPON  A  PREVIOUS 
KNOWLEDGE  OF  THE  SUBJECTS  TREATED. 

THREE-PHASE  CIRCUITS 

The  copper  required  to  install  the  complete  circuit 
between  generator  or  transformer  and  its  load  is  a 
considerable  item  in  the  cost  of  electrical  power  trans- 
mission projects. 


OUT  OF  PHASE  TORQUES 


59 


Let  us  take  three  transformers  with  a  load  for  each 
and  attempt  to  arrange  them  with  respect  to  the  least 


vw      Uvv      UM 


iws/      tvw      Uvv 


FIG.  29 

possible  outlay  of  copper.    The  elementary  installa- 
tion with  three  separate  circuits  is  shown  in  Fig.  29, 


60 


CHAPTER   VI 


where  A ,  B,  C  represent  the  transformers,  D,  E,  F  their 
respective  loads,  and  1,  2,  3,  4,  5,  6  the  copper  wires 
composing  the  circuits. 

In  Fig.   29b,  with  proper  regard  for  the^current 
directions,  wire  number  4  has  been  omitted. 


FIG.  30 

In  Fig.  29,  a  similar  consolidation  of  circuits  B — E 
and  C — F  has  been  effected. 

In  Fig.  29c,  it  will  be  seen  that  wires  1  and  6  are 
the  only  ones  having  but  one  connection  to  load  and 
transformer,  and  Fig.  29d  shows  a  consolidation 
involving  the  torque  of  all  three  transformers,  making 
the  circuit  of  three  wires  the  equivalent  of  six. 

The  torques  of  the  three  transformers  are  adjusted 
to  follow  one  another  at  120°  intervals  by  the  dis- 
tribution of  conductors  in  the  generator,  as  shown 
by  the  sketch,  Fig.  30. 


OUT   OF   PHASE   TORQUES 


61 


The  generator  coils,  or  phase  windings  as  they  are 
called,  are  indicated  by  A,  B,  and  C.  Their  electro- 
motive forces  are  represented  by  A,  B,  C  of  Fig.  31. 
The  field  marked  N — S  in  Fig.  30  revolves  inside  the 


FIG.  31 

stationary  armature,  thus  allowing  the  load  currents 
to  be  taken  from  A ,  B,  C  without  moving  contacts. 
The  electromotive  forces  generated  in  the  position 
shown  in  the  sketch  of  the  alternator  circuits  may  be 
found  Jfrom  Fig.  31  at  the  30°  and  210°  points;  A 
and  C  are  at  one-half  maximum  in  one  direction, 
while  B  is  at  its  maximum  in  the  opposite  direction. 


62  CHAPTER  VI 

It  will  be  noted  that  at  every  instant  the  sum  of 
the  three  voltages,  taking  their  direction  into  account, 
is  zero.  Consequently  there  will  not  be  any  danger  in 
short-circuiting  the  three  transformers,  as  shown  in 
Fig.  30.  This  connection  is  known  as  the  delta. 

The  student  will  find  it  convenient  to  reproduce 
Fig.  31,  and  draw  vertical  lines  through  the  degree 
points  in  the  figure.  These  lines,  wherever  drawn, 
will  show  the  amount  and  direction  of  the  voltage  in 
each  of  the  alternator  circuits. 

Connection  of  alternator  and  transformers  is 
shown  in  Fig.  32. 

Three-phase  connections  are  also  made  in  the 
form  known  as  the  Y  or  star  connection.  The  evolu- 
tion of  this  type  of  connection  is  shown  in  Fig.  33. 

Simple  internal  circuits  of  a  transformer  are  shown 
in  Fig.  34. 

As  was  previously  stated,  the  power  in  a  single- 
phase    alternating-current    circuit   is    given    by    the 
expression  El  cos  0,  where 
E  =  effective  volts; 
I  =  effective  amperes; 
6   =  lag  angle; 

cos  6   —  power  factor. 

From  inspection  of  Fig.  32,  it  will  be  seen  that 
the  line  voltage  per  phase  in  the  delta  system  is  the 
same  as  the  voltage  per  phase  in  the  alternator. 
The  current  per  line,  however,  is  not  equal  to  the 
current  per  alternator  phase,  but  will  be  found  from 
Fig.  31  to  be,  making  proper  allowance  for  sign, 
IL  =  V/3  /, 


OUT  OF   PHASE  TORQUES  63 


FIG.  32 


64 


CHAPTER   VI 


Where 

IL  =  line  current, 
and 

I tt  =  current  in  the  alternator  phase, 
effective  values  being  used. 


a 


>v 


FIG.  33 

Similarly  in  the  Y  or  star  system  of  connection, 
the  current  is  seen  to  be  the  same  in  line  and  alter- 
nator phase,  but  the  voltage  of  the  line  is 

EL  =  \/3~X 
where 

EL  =  line  voltage, 


OUT  OF   PHASE  TORQUES 


65 


TO 


TO  LOAO 


FIG.  34 


66  CHAPTER  VI 

and 


,  =  phase  voltage 


effective  values  being  used. 

Now  in  either  system  if  the  load  is  evenly  dis- 
tributed among  the  three  phases,  the  power  will  be 
equal  to 

P  =  3  E,  /0cos  0 

where  P  —  power,  and  other  symbols  have  meanings 
given  above.  But  in  practice  the  line  values  and  not 
the  phase  values  are  ordinarily  known. 

Hence  from  EL  =  V  3  £,,  above, 


and 

P  =  3^7,  cos  0 
V3 

In  the  star  system 

1 1  ==  IL 
and 

P  =  V  3  ELIL  cos  < 

For  the  delta  system 

IL  =  V/T/, 

EL  =  E* 
and 

P  =  3  Et-£  cos  0    =  3  E 
or,  in  both  systems, 

P   =    V/3EI/LCOS  ( 


OUT   OF   PHASE   TORQUES 


67 


Power  is  measured  in  practice  by  the  use  of  watt- 
meters. Single-phase  power  may  be  measured  by 
inserting  the  current  coil  of  a  wattmeter  in  one  line, 
while  the  potential  coil  is  connected  across  the  line. 

Three-phase  power  may  be  measured  with  two  watt- 
meters connected  as  shown  in  Fig.  35. 


FIG.  35 


The  sum  of  the  readings  of  the  meters  will  give 
the  total  power  when  the  lag  angle  is  less  than  30°, 
and  their  difference  when  the  angle  is  greater  than 
30°. 

The  proof  of  the  above  follows: 

P  =  total  power  at  any  moment; 

0i,  02,  03,  and  ii,  ii,  is,  are  the  phase  voltages  and 
currents. 

P  =  dii  +  e«#2  +  03*3  (instantaneous); 

ii  +  it  +  i*  =  0  (see  Fig.  31); 
whence 

*2  =  —  (ii  +  is) ; 


68  CHAPTER   VI 

and  substituting  above, 

P  =  d  ii  ez  (ii  +  is)  +  ez  t3; 

P  =  (ei  —  ez)  ii  +  4  Oa  —  e2). 
Now  ei  —  ez  can  be  shown  to  be  the  voltage  across 
the  potential  coil  of  meter  A,  and  ez  —  ez  is  the  voltage 
across  the  potential  coil  of  meter  B. 

Also  (ei  —  62)  i\  is  proportional  to  the  torque 
exerted  on  the  moving  element  of  meter  A,  and 
(03  —  62)  ia  is  proportional  to  the  torque  exerted  on 
the  moving  element  of  meter  B;  and  hence  by  proper 
calibration  of  scale  the  sum  of  A  and  B  will  represent 
the  total  average  power  of  the  three  phases,  for  the 
inertia  of  the  meter  pointers  will  automatically 
average  the  instantaneous  values  and  will  therefore 
read  average  power. 


CHAPTER  VII 

The  Condenser — Functions;  analysis  of  condenser  action;  the 
dielectric;  current  and  energy  relations;  practical  example. 


STUDENT'S  GUIDE 

From  the  study  of  induction  and  its  effects  we  pass 
to  the  consideration  of  the  condenser,  an  apparatus 
which  has  operating  characteristics  exactly  opposite 
to  those  of  induction,  but  which,  when  existing  by 
themselves  in  excess,  are  undesirable.  The  neutraliz- 
ing effect  of  capacity  and  inductance  is  explained,  and 
a  practical  circuit  containing  both  is  discussed. 

THE  CONDENSER 

A  condenser  is  usually  described  as  consisting  of  two 
insulated  conductors  charged,  one  with  positive,  and 
the  other  with  negative  electricity.  If  the  two  charged 
conductors  are  connected,  either  by  contact  with  each 
other  or  through  a  third  conductor,  a  current  passes 
which  restores  the  electric  equilibrium.  The  charged 
conductors  are  said  to  be  capable  of  containing  a 
certain  quantity  of  electricity.  The  condenser  is 
said  to  have  capacity.  In  the  application  of  our 
analogy,  we  have  no  occasion  to  recognize  electricity 
as  an  entity,  and  therefore  we  must  inquire  how 
the  analogy  explains  the  phenomena  of  the  electric 
condenser. 

Our  analogy  regards  a  conductor  as  containing  an 
indefinite  number  of  infinitesimal  shafts,  "reasonably 


70 


CHAPTER   VII 


rigid  and  elavStic  with  regard  to  torsion."  Under  a 
torsional  stress,  these  infinitesimal  shafts  will  have  a 
torsional  strain,  and  being  elastic,  they  will  return  to 
their  normal  condition  unless  prevented  by  the 
contact  of  a  non-conductor.  We  therefore  define  a 
charged  condenser  as  two  insulated  conductors  whose 
lines  of  elementary  flexible  shafts  are  subject  to 


FIG.  36 

torque.     The  material  used  to  insulate  the  condenser 
plates  or  conductors  is  termed  a  dielectric. 

In  Fig.  36,  let  abcde  be  a  simple  circuit  with  the 
condenser  composed  of  two  copper  plates  A ,  A ,  and 
the  single  plate  B,  the  connections  (except  a,  g,  /) 
being  as  shown.  If  the  condenser  were  removed  and 
the  circuit  completed  by  joining  the  wires  a  and  /, 
the  maximum  voltage  in  the  circuit  with  a  given 
current  would  be  that  due  to  the  resistance  of  the 


THE   CONDENSER  71 

circuit.  When  the  condenser  is  in  the  line  (as  in  the 
figure,  omitting  a,  g,  f)  the  insulation  between  the 
plates  A,  A,  and  the  plate  B  prevents  a  current 
passing  between  the  plates.  If  an  alternating  elec- 
tromotive force  be  applied  at  c,  it  will  be  trans- 
mitted to  the  plates  A,  A,  and  plate  B.  If  the 
transmitting  elementary  shafts  in  the  conductor  were 
absolutely  rigid,  there  could  be  no  current  in  any 
part  of  the  line;  but  as  the  elementary  shafts  are 
subject  to  torsional  strain,  and  as  they  are  elastic,  all 
the  elementary  shafts  in  the  plates  will  be  subjected 
to  a  torsional  alternating  strain.  This  strain  will  be 
the  greater  because  the  insulation  of  the  plates  will 
permit  a  greater  voltage  in  the  circuit  than  was 
possible  when  the  condenser  was  not  in  the  line. 

The  torsional  strain  in  the  elastic  shafts  represents 
a  condition  of  energy,  which  energy  will  be  released 
when  the  strain  is  relieved.  When  the  alternating 
voltage  has  been  applied  for  a  time  at  c,  if  the  wires 
are  disconnected  at  a  and  /,  the  condenser  will  be 
"charged."  In  this  state  it  is  said  to  contain  a 
certain  quantity  of  electricity.  If  connection  is  made 
between  the  plates,  say  by  a  conductor  a,  g,  /,  there 
will  be  a  momentary  current  in  this  conductor.  We 
attribute  this  to  the  release  of  the  stress  on  the 
elementary  shafts,  thus  allowing  them  by  their 
elasticity  to  give  up  their  stored  energy  due  to  the 
torsional  strain  to  which  they  were  subjected  so  long 
as  they  were  insulated. 

The  operation  of  the  condenser  under  the  applica- 
tion of  an  alternating  e.m.f.  is  to  bring  the  torque  in 


72  CHAPTER  VII 

the  elementary  shafts  gradually  to  a  maximum  when 
the  applied  e.m.f.  is  a  maximum.  This  is  90°  from 
the  zero  of  the  e.m.f.  At  this  point  the  e.m.f.  begins 
to  diminish  and  the  energy  stored  in  the  shafts  due  to 
their  torsional  strain  is  gradually  given  off  and  the 
strain  becomes  zero  again  when  the  e.m.f.  becomes 
zero.  This  process  is  repeated  during  the  next  180°, 
but  by  the  rotation  of  the  shafts  in  the  opposite 
direction.  In  long  lines  the  resistance  lowers  the 
voltage.  A  condenser  on  the  end  of  the  line  permits 
the  voltage  to  be  increased  even  to  distant  parts  of 
the  line.  In  the  many  elementary  shafts  in  the  plates 
of  the  condenser,  energy  is  alternately  stored  and  given 
off,  and  with  increased  voltage,  as  above  described. 

In  our  analogy  the  elementary  flexible  shafts  in  the 
dielectric  are  capable  of  receiving  and  transmitting  an 
electromotive  force  (or  torque)  without  evidence  of 
current  in  the  dielectric.  In  a  dielectric,  as  well  as 
in  a  conductor,  the  properties  of  the  elementary 
flexible  shafts  must  vary  in  different  substances, 
particularly  in  reference  to  their  rigidity,  their 
elasticity,  and  their  resistance. 

If  the  elementary  shafts  are  quite  rigid  in  regard 
to  torsion,  they  will  have  a  comparatively  small  tor- 
sional strain  under  a  given  e.m.f.  If  at  the  same 
time  these  elementary  shafts  have  low  elasticity, 
they  will  give  back  but  a  small  part  of  the  energy 
required  to  produce  in  them  a  given  torsional  strain. 
If  the  elementary  shafts  have  great  resistance  due  to 
their  mutual  interference,  a  slight  rotary  motion  or 
displacement  will  cause  the  resistance  to  balance  the 


THE   CONDENSER  73 

applied  e.m.f.  This  rotary  motion  may  be  so  small 
as  not  to  be  ordinarily  measurable  as  current. 

Following  our  analogy,  we  must  consider  the  lines 
of  molecules  in  the  dielectric  (considering  air  for  the 
present)  as  subject  to  torque.  Air  is  not  a  complete 
non-conductor.  When  the  e.m.f.  is  sufficient,  current 
passes,  always  accompanied  by  an  appearance  of  heat. 
It  may  be  reasonably  assumed  that  a  very  limited 
rotary  motion  of  these  elementary  shafts  takes  place 
before  the  current  is  noticeable.  If  the  condenser 
plates  are  very  widely  separated,  the  condenser  effect 
is  neutralized  or  greatly  diminished.  This  is  explained 
by  considering  that  the  resistance  to  the  slight  rota^ 
tion  of  the  lines  of  molecules  in  the  dielectric  (which 
extend  far  out  into  the  atmosphere);  the  torsional 
strain,  and  the  low  elasticity  of  these  elementary 
shafts,  all  combine  to  absorb  most  of  the  energy  sup- 
plied to  the  condenser.  Therefore,  when  the  plates 
are  separated,  the  energy  is  not  stored  in  the  elemen- 
tary shafts  of  the  condenser;  in  other  words,  the  con- 
denser cannot  become  "charged." 

When  the  condenser  plates  are  brought  closer  to- 
gether, each  short  molecular  shaft  in  the  dielectric 
has  a  torque  applied  to  each  of  its  ends,  and  in  the 
same  direction ;  and  while  this  short  shaft  rotates  until 
the  resistance  of  the  dielectric  balances  the  torque 
applied  to  it,  there  is  but  little  energy  absorbed  in 
the  dielectric,  because  the  shafts  between  the  plates 
of  the  condenser  are  so  short. 

If  different  substances  are  used  for  the  dielectric, 
the  efficiency  of  the  condenser  will  vary  with  the 


74  CHAPTER   VII 

character  of  these  different  substances.  Dielectrics 
for  condensers  are  classified  with  reference  to  their 
several  specific  inductive  capacities.  The  greater  the 
specific  inductive  capacity  of  the  dielectric,  the  greater 
the  efficiency  of  the  condenser. 

Air  has  a  less  specific  inductive  capacity  than  glass; 
that  is,  air  is  a  better  conductor  of  electricity  than 


glass,  and  therefore  absorbs  more  of  the  applied  energy 
than  glass  when  used  as  the  dielectric  of  a  condenser. 

It  is  a  matter  of  speculation  how  far  the  stress  in  the 
elementary  shafts  in  the  air  may  extend,  and  at  what 
distances  these  lines  may  have  slight  rotary  motion. 
It  does  not  seem  an  unreasonable  stretch  of  imagina- 
tion to  suggest  that  there  may  be  enough  motion  at 
great  distances  to  transmit  wireless  messages. 

The  graphical  representation  of  e.m.f.  current,  and 
energy  in  connection  with  the  electric  condenser, 
follows. 

We  shall  take  a  sine-wave  impressed  voltage  and 
attempt  to  find  the  logical  current  result. 


THE   CONDENSER  75 

If  our  condenser  is  uncharged,  and  if  the  electro- 
motive force  be  applied  just  as  it  is  growing  from  zero 
to  a  positive  maximum,  at  A,  Fig.  37,  a  stress  is  placed 
upon  the  elastic  molecular  filaments  of  the  condenser. 
This  stress  is  resisted  by  the  dielectric.  Consequently 
a  strain  results  in  the  molecular  filaments  of  the  con- 


FIG.  38 

denser  which  appears  in  the  leads  of  the  condenser  as 
current.  This  current  will  increase  until  the  torque 
reaches  its  maximum  B,  and  will  then  commence  to 
diminish,  when  the  condenser  filaments,  being  re- 
lieved of  their  maximum  stress,  will  begin  to  rotate  in 
the  opposite  direction  at  C,  where  there  is  a  reversal 
of  current  direction,  as  shown.  The  current  persists 
while  the  stress  remains,  but  becomes  zero  at  F  when 
the  torque  dies  to  zero.  From  that  point  the  action 
repeats,  with  FDE  corresponding  to  AB  and  C. 


76  CHAPTER  VII 

The  curve  of  energy  is  shown  in  Fig.  38.  Now  if  we 
construct  a  similar  diagram  for  a  circuit  containing 
pure  inductance,  we  shall  find  a  clue  to  the  means  by 
which  capacity  offsets  the  effect  of  inductance. 

In  Fig.  39,  the  current  C  is  90°  behind  the  position 
it  would  occupy  if  the  circuit  were  non-inductive. 


FIG.  39 

The  product  of  C  by  E  is  energy,  and  produces  curve 
F,  which  will  be  seen  to  be  180°  displaced  from  the 
energy  curve  of  capacity,  Fig.  38.  Hence  we  may 
readily  understand  the  opposite  effects  of  capacity 
and  inductance. 

The  following  example  will  serve  to  give  a  better 
conception  of  the  practical  side  of  the  matter. 

If  we  impress  an  alternating  voltage  £  on  a  circuit 
consisting  of  a  non-inductive  resistance  R,  a  certain 
current  Cwill  result,  and  the  rate  of  energy  transmis- 
sion we  may  call  P.  This  is  represented  by  Fig.  40. 


THE   CONDENSER 


77 


Let  us  now  introduce  an  inductance  5  into  the  cir- 
cuit, as  shown  in  Fig.  41.  C  will  be  decreased,  and  P 
will  be  decreased.  The  inductance  has  introduced  a 
torque  of  self-induction  which  has  opposed  the  trans- 
mission of  energy. 


FIG.  40 


FIG.  42 

If  now  we  insert  in  the  circuit  a  condenser  with  a 
variable  number  of  plates,  Fig.  42,  we  shall  find  the 
current  C  and  the  power  P  slightly  increased. 

By  increasing  the  number  or  area  of  the  plates  in  the 
condenser,  the  current  and  power  can  be  increased 
to  very  nearly  the  values  they  had  before. 


CHAPTER  VIII 

Measuring  Instruments — Principles  of  direct-current  meters; 
calculation  of  meter  constants  for  use  of  shunts  and  multi- 
pliers; wattmeters.  Alternating-Current  Meters  —  Dyna- 
mometer and  hot-wire  type.  Resistance  Measurements — 
Volt-ammeter  method;  Wheatstone's  bridge.  The  Magneto. 
Units  of  Electrical  Measurement. 


STUDENT'S  GUIDE 

In  this  chapter  we  study  the  instruments  used  to 
measure  voltage,  current,  resistance,  and  power. 
Numerical  examples  are  given  to  show  the  method 
of  constructing  meters  which  will  be  able  to  measure 
within  predetermined  limits.  A  table  of  electrical 
units  is  also  given. 

MEASURING  INSTRUMENTS 

Reference  has  been  made  in  preceding  chapters  to 
various  measuring  instruments  the  characteristics  of 
which  will  be  outlined  here.  Voltage,  current,  and 
power  are  the  three  quantities  which  we  should  be 
able  to  measure  if  we  are  to  be  certain  that  a  given 
installation  is  doing  the  work  expected  or  of  which  it 
is  capable. 

The  direct-current  voltmeter  and  ammeter  are 
interchangeable,  provided  shunts  and  multipliers  are 
furnished.  The  meter  itself  consists  of  a  small  moving 
coil  suspended,  on  spiral  springs  and  jewel  bearings, 
in  the  air  gap  between  the  pole-faces  of  a  strong 


DIRECT 

CURRENT 

METERS 


SHUNT 


MEASURING   INSTRUMENTS  79 

permanent  magnet.  The  coil  carries  a  pointer  which 
moves  over  a  scale  when  the  coil  rotates.  Rotation 
of  the  coil  is  produced  by  motor  action  when  a  current 
passes  through  its  turns,  reacting  with  the  field  of  the 
permanent  magnet.  The  turning  moment  is  propor- 
tional to  the  current  in  the  coil,  and  rotation  is 
restrained  by  the  spiral  springs  with  a  force  propor- 
tional to  divsplacement ;  hence  the  pointer  moves  over 
a  distance  proportional  to  the  strength  of  the  current 
in  the  moving  coil. 

For  very  small  currents  the  meter  as  described  is 
satisfactory;  but  when  larger  currents  are  carried  in 
and  out  of  the  coil,  the  heating  effect  on  the  springs 
which  act  as  carriers  for  the  current,  causes  errors, 
and  the  heating  limit  of  the  turns  themselves  is  soon 
reached.  Consequently  shunts  are  always  used,  so 
that  only  a  minute  part  of  the  total  current  passes 
through  the  coil.  The  scale  of  the  meter  is  calibrated 
on  the  principle  of  the  division  of  current ;  that  is,  the 
current  carried  by  the  shunt  and  the  coil  are  in  the 
inverse  ratio  of  their  respective  resistances. 

The  same  meter  without  the  shunt  is  used  as  a  volt- 
meter by  inserting  a  large  series  resistance.  Com- 
mercial meters  are  usually  made  so  that  full  scale 
deflection  is  obtained  with  the  passage  of  0.01  ampere 
of  current.  Consequently  if  the  resistance  of  the  meter 
is  known,  the  resistance  which  must  be  placed  in  series 
with  it  to  limit  the  current  to  0.01  ampere  can  easily 
be  calculated. 

For  instance,  let  us  calculate  the  resistance  neces- 
sary in  the  shunt  and  "multiplier,"  as  the  series 


80  CHAPTER   VIII 

resistance  is  called,  for  a  meter  which  is  to  read  a 
maximum  of  50  amperes  and  250  volts.  Reasonable 
assumptions  would  be  a  meter  resistance  of  60  ohms, 
and  a  meter  current  of  0.01  ampere  for  full  scale 
deflection. 

To  find  the  shunt  resistance:  the  current  carried 
by  the  shunt  will  be  49.99,  and  by  the  meter  0.01, 
when  the  pointer  is  at  the  top  of  the  scale.  The 
resistances  must  be  inversely  proportional  to  these 
values.  Let  R8  represent  shunt  resistance. 

R8        0.01 
#M~"49.99 

0  01 

R8  =  —      -  X  60  =  0.012  ohms 
49.99 

The  shunt  we  should  expect,  therefore,  to  be  a  short 
piece  of  metal  of  large  cross-section,  and  this  is  in  fact 
the  form  used.  The  value  of  resistance  indicated  as 
necessary  is  more  easily  arrived  at  by  trial  and  com- 
parison with  standard  instruments  than  by  construc- 
tion to  exact  dimensions,  and  that  procedure  is  usually 
followed. 

The  calculation  of  the  series  resistance  needed  to 
construct  a  multiplier  for  the  case  assumed  above  is  as 
follows:  The  voltage  to  be  impressed  when  the  full 
scale  deflection  is  caused  is  250;  the  current  flowing 
must  be  0.01  ampere.  By  Ohm's  Law  the  total  re- 

•p         2C?0 

sistance  =  R  =  -  =  =  25,000  ohms.  The  volt- 
meter itself  has  only  60  ohms  in  its  coil,  so  that  25,000 
—  60  =  24,940  ohms  must  be  placed  in  the  multiplier. 


MEASURING   INSTRUMENTS  81 

Multipliers  are  made  by  winding  great  lengths  of  fine 
wire  on  cards  which  are  mounted  in  a  ventilated  box. 

When  a  meter  is  wanted  to  measure  power,  one  hav- 
ing the  characteristics  of  both  the  ammeter  and  volt- 
meter is  necessary,  for  the  deflection  of  the  pointer 
must  be  proportional  to  the  product  of  volts  and 
amperes. 

To  accomplish  this  the  permanent  magnet  is  re- 
placed by  a  second,  stationary  coil  of  wire.  If  we 
cause  a  current  to  flow  in  the  stationary  coil  propor- 
tional to  the  current  in  the  circuit  in  which  we  desire 
to  measure  the  power,  and  at  the  same  time  cause  a 
current  proportional  to  the  voltage  of  that  circuit  to 
flow  in  the  moving  coil  of  our  instrument,  there  will 
be  produced  a  motor  action  as  in  the  voltmeter  and 
ammeter;  but  the  deflection  will  be  proportional  to 
the  product  of  two  variables — that  is,  to  the  product 
of  E  and  C,  or  proportional  to  power.  Such  an  instru- 
ment is  called  a  wattmeter.  Current  for  the  stationary 
coil  is  obtained  as  in  the  ammeter,  and  for  the  moving 
coil  as  in  the  voltmeter. 

The  ammeter  and  voltmeter  described  above  can 
be  used  only  on  direct  current,  but  the  wattmeter  will 
read  alternating-current  power,  because  the  direction 
of  the  field  caused  by  the  stationary  coil  changes 
when  the  direction  of  the  current  in  the  moving  coil 
changes,  and  consequently  the  rotating  force  is  con- 
stant in  direction ;  and  while  it  varies  with  the  growth 
and  decrease  of  the  current  and  voltage,  the  pointer 
will  take  up  a  position  which  indicates  the  average 
power. 


82  CHAPTER   VIII 

Alternating-current  ammeters  and  voltmeters  are 
made  with  two  coils  in  series,  in  order  that  the  reversal 
of  current  may  not  reverse  the  force  causing  deflection 
of  the  meter  pointer.  If  reference  is  made  to  the  "left- 
hand  rule"  stated  in  Chapter  V,  it  will  be  seen  that  if 
both  flux  and  current  direction  are  changed,  the  force 
exerted  on  the  conductor  will  be  unchanged.  These 
are  called  dynamometer  type  instruments. 

Alternating-current  ammeters  are  also  made  on  the 
principle  of  the  heat  developed  in  a  wire  carrying  cur- 
rent, the  wire  expanding  when  heated,  thus  allowing 
a  spring-controlled  pointer  to  move  over  a  scale  cali- 
brated to  read  amperes. 

Illustrations  shown  are  from  the  instrument  catalog 
of  the  Wagner  Electric  Manufacturing  Company. 

In  order  to  measure  resistance,  methods  depending 
more  or  less  directly  on  Ohm's  Law  are  used.  The 
voltmeter-ammeter  method  of  resistance  measurement 
applies  the  law  directly  by  calculating  resistance  from 
simultaneous  readings  of  current  and  voltage  drop 
across  the  resistance  in  question. 

Wheatstone's  bridge  is  an  apparatus  much  used  for 
resistance  measurements.  It  consists  of  means  for 
balancing  known  resistances  against  the  unknown 
which  it  is  desired  to  measure.  Kelvin's  double  bridge 
is  a  modification  of  the  Wheatstone  circuit  by  means 
of  which  very  low  resistances  may  be  accurately 
measured. 

The  magneto  is  a  small  alternating-current  generator 
with  a  permanent  magnet  field  and  two-pole  revolving 
armature.  The  voltage  generated  by  the  ordinary 


MEASURING  INSTRUMENTS  83 

hand-driven  testing  magneto  is  comparatively  high, 
but  the  current  output  is  negligible. 

Illustrations  of  meters  and  parts  are  from  the  instru- 
ment catalog  of  the  Wagner  Electric  Manufacturing 
Company. 

UNITS  OF  ELECTRICAL  MEASUREMENT 

The  following  list  of  electrical  quantities  and  names 
of  units  is  taken  from  the  standardization  rules  of  the 
American  Institute  of  Electrical  Engineers. 

Capacity,  farad 

Current,  ampere 

Dielectric  constant,  — (a  number) 

Electromotive  force,  volt 

Energy,  joule  or  watt-hour 

Frequency,  cycle  per  second 

Impedance,  ohm 

Inductance,  henry 

Magnetic  flux,  maxwell 

Permeability, 

Phase,  degree 

Power,  watt 

Quantity  of  electricity,  coulomb 

Reactance,  ohm 

Resistance,  ohm 

Since  we  have  defined  electricity  as  a  means  of 

transmitting  energy,  two  of  the  terms  in  the  list  given 

above  are  obviously  misnomers;  they  are  "quantity" 

and  "capacity."    Quantity  may  be  discarded  entirely, 

while  capacity  will  serve,  failing  to  find  a  better  word, 


84  CHAPTER   VIII 

to  describe  the  property  of  the  condenser  by  virtue  of 
which  it  neutralizes  the  effect  of  inductance.  Capacity 
is  calculated  from  the  expression 

r  AK 

Capacity  =  — 

where  A  =  area  of  active  plates,  D  is  thickness  of 
dielectric,  and  K  is  a  constant  depending  on  the 
material  of  the  dielectric.  The  rational  unit  becomes 

Area 

—  =  Length 

Length 

that  is,  inches  or  centimeters. 

Current  we  have  already  defined  as  angular  velocity. 

Dielectric  constant  is  a  numeral. 

Electromotive  force  or  voltage  has  been  assumed  to 
be  torque. 

Energy  may  be  expressed  as  horsepower  hours. 

Frequency  is  a  number. 

Impedance  is  the  resultant  resistance  when  a  circuit 
has  inductive  characteristics. 

Inductance  is  a  magnetic  quality. 

Magnetic  flux — imaginary  lines  indicating  the 
direction  of  forces  acting  in  the  space  about  a  current 
and  in  the  neighborhood  of  magnets. 

Permeability  is  a  number. 

Phase  is  the  angular  relation  between  various 
torques  and  rotating  filaments. 

Power  is  the  rate  of  doing  work  and  is  expressed  in 
horsepower. 

Reactance  is  the  component  of  impedance  intro- 
duced by  inductance. 


MEASURING  INSTRUMENTS  85 

Resistance  is  the  counter  effort  made  by  various 
elements  in  an  electric  circuit  to  prevent  rotation  of 
molecular  filaments. 
The  revised  list  then  is: 

Capacity,  a  number 

Current,  r.p.m. 

Dielectric  current,  a  number 

Electromotive  force,  foot  pounds  of  torque 

Energy,  horsepower  hours 

Frequency,  cycles  per  second 

Impedance,  a  number 

Inductance,  (magnetic) 

Magnetic  flux,  (magnetic) 

Permeability,  (magnetic) 

Phase,  degrees 

Power,  horsepower 

Reactance,  a  number 

Resistance,  a  number 


CHAPTER  IX 

General  Remarks  and  Recapitulation. 

If  the  analogy,  as  described  in  this  work,  were 
adopted  as  the  basis  for  illustrating  and  explain- 
ing the  various  phenomena  connected  with  the 
electric  transmission  of  power,  we  believe  it  can  be 
demonstrated  that  many  of  the  difficulties  which 
the  student  encounters  would  be  removed.  It  does 
not  seem  to  the  writer  possible  for  one  to  under- 
stand or  explain  how  a  small  copper  wire  can  convey 
a  hundred  or  a  thousand  horsepower  of  energy  by 
means  of  pulsations  back  and  forth  in  or  along  or 
around  the  conductor.  We  have  no  experience 
with  any  such  conditions  producing  similar  results. 
The  student  may  try  to  accept  what  he  is  told  and 
often  tries  in  vain.  The  analogy  herein  suggested 
deals  with  something  familiar.  The  ordinary  pro- 
cesses of  reasoning  lead  to  results  which  agree  in 
nearly,  if  not  all,  cases  with  known  and  accepted 
facts,  established  by  inductive  study  and  experience. 
For  the  time  at  least,  all  reference  to  such  terms  as 
electric  energy  as  related  to  transmission  of  energy 
by  electricity  could  be  dropped.  The  analogy  does 
not  call  for  any  attempt  to  define  electricity  as 
something  that  exists,  but  only  as  a  means  of  carry- 
ing energy.  It  translates  the  symbols  of  the  basic 
law  known  as  Ohm's  Law  into  the  common  language 
of  mechanics — namely,  torque  and  angular  velocity. 


RECAPITULATION  87 

It  does  not  limit  the  subject  to  any  of  these  element- 
ary ideas.  It  leaves  plenty  of  room  for  imagina- 
tion and  investigation  into  ultimate  realities  and 
processes.  But  it  leads  to  practical  results  by  simple 
methods  of  reasoning.  It  brings  ready  insight  into 
otherwise  difficult  problems.  In  a  word,  it  permits 
the  use  of  the  deductive  method  of  approach  by 
quick  and  easy  paths  to  a  goal  which  it  has  taken 
many  years  to  reach  by  the  inductive  method. 

The  foregoing  is  an  outline  of  the  ready  applica- 
tion of  our  analogy  and  its  accompanying  working 
hypothesis.  It  has  shown,  it  seems  to  us,  some 
remarkable  results;  results  that  are  suggestive  of 
the  value  and  scope  of  the  new  approach  to  the 
study  of  electricity. 

Some  of  these  results  are  summarized  as  follows: 

1.  The  analogy  leads  to  Ohm's  Law  by  deduc- 
tion. 

2.  It    translates  the  symbols  and  statements  of 
that  law  into  well  understood  mechanical   terms — 
namely,    torque,    angular   velocity,    and    energy   or 
work. 

3.  In   connection   with    the   working   hypothesis 
it  shows  what   may  be   the  process  of  generating 
electricity  in  the  dynamo  and  why  the  current  is 
alternating,  as  generated. 

4.  It  reveals  the  use  and  operation  of  the  com- 
mutator. 

5.  It    explains    induction    and    shows    that    the 
induced  current  is  90°  behind  the  primary  current. 

6.  It  predicts  and  explains  the  transformer. 


88  CHAPTER  IX 

7.  It  shows  a  ready  application  to  the  operation 
and  effect  of  the  condenser. 

8.  It  shows  how  the  effect  of  two  out  of  phase 
torques,    when    combined,   causes   what  is    usually 
termed  lag,  and  how  to  plot  and  measure  the  lag. 

9.  It   indicates   how   an  interrupted  direct  cur- 
rent in  the  primary  coil  of  a  transformer  may  give 
an  alternating  current  in  the  secondary  coil. 


CHAPTER  X 

The  Solution  of  Problems. 

With  a  given  circuit,  certain  results  are  invariably 
attained  by  successive  applications  of  the  same  elec- 
tromotive force,  and  by  making  use  of  certain  rules 
and  formulae  it  is  possible  to  predict  what  will  happen 
in  any  given  case.  The  following  group  of  problems 
will  illustrate  some  types  of  circuits,  and  the  prin- 
ciples employed  may  be  generally  applied. 

PROBLEM  1 

To  find  the  current  when  a  known  voltage  is 
impressed  on  a  combination  of  resistances. 

Discussion.  (See  definition  of  electric  circuit, 
Chapter  III.) 

Resistances  are  said  to  be  in  series  or  parallel 
according  to  whether  the  rotating  filaments  are 
all  contained  in  one  conductor  or  divided  among 
two  or  more.  When  the  whole  current  of  a  line 
is  sustained  at  once  in  two  or  more  resistances, 
those  resistances  are  said  to  be  in  series;  but  when 
the  filaments  divide  at  one  junction,  part  extend- 
ing through  one  resistance  and  the  rest  through 
others,  uniting  at  a  second  junction  and  forming 
one  body  thence  through  the  line,  the  resistances 
are  said  to  be  in  parallel. 

Resistances  in  series  are  added  to  find  the  total 
or  equivalent  resistance.  The  equivalent  or  total 


90  CHAPTER   X 

resistance  of  several  resistances  in  parallel  may  be 
found  as  in  Fig.  43. 

In  the  circuit  shown,  current  in  branch  a  is,  by 
Ohm's  Law, 


In  branch  b 


^       12 

C  =  —   =6  amperes 

JL 


n         12 

C  —  —  =  4  amperes 
3 


12V 


OAm* 


The  line  current  therefore  is 

IL  —  6  +  4  =  10  amperes 

The  equivalent  resistance  R0  is  that  resistance  with 
which  a  and  b  might  be  replaced  without  changing  the 
value  of  IL. 

From  Ohm's  Law 


12 

RL  =  —  =  1.2  ohms 
10 


PROBLEMS  91 


We  may  also  write 

E    _E         E 

TT>  ^j  I       13/1 

./Vfl  -*MI  J\& 

or 


and  in  general 


l  =1  +1 

R       R,      R, 


i        1+i    +1+... 

R  Rl  XV2  -^-3 


Using  the  general  equation,  the  student  should 
check  the  value  of  RL  given  above. 

PROBLEM  2 

A  number  of  "exit"  signs  and  fire-alarm  boxes 
are  to  be  illuminated  with  low-voltage  lamps  run  from 
storage  batteries,  the  purpose  being  to  have  a  source 
of  power  entirely  unaffected  by  any  accident  to  the 
regular  lighting  circuits  of  the  building. 

The  lamps  available  are  6-volt  2-ampere  size,  and 
the  storage  battery  consists  of  a  number  of  lead 
cells  of  32  ampere-hour  capacity.  (See  Chapter  III.) 

If  20  lamps  are  to  be  used  11  hours  a  day,  how 
many  storage  cells  will  be  needed  and  how  should 
they  be  arranged,  and  how  connected  to  the  lamps? 

Solution.  The  lamps  should  be  connected  in 
parallel  in  groups  as  indicated  in  Fig.  44.  If  placed  in 
series  a  higher  voltage  would  be  required,  and  the 
burning  out  of  one  lamp  would  put  all  out  of  service. 
The  groups  are  made  as  small  as  convenient  to  avoid 
the  large  line  drop  of  voltage  and  consequent  dimming 


92 


CHAPTER   X 


of  lamps  at  the  end  of  the  line,  which  would  be  un- 
avoidable if  one  long  parallel  row  were  used. 

Three  lead  cells  in  series  are  known  as  a  nominal 
6-volt  battery. 


[ 

1 

; 

kD- 

0 

•o 

0 

O 

o 

o 

o- 

o- 

rO 

o 

0- 

O- 

o 

o 

o 

k> 

k>J 

k> 

-QJ 

LAMP6 


FIG.  44 


Twenty  of  these  lamps  require  40  amperes. 

From  Chapter  III  we  find  that  when  a  lead  cell  is 
rated  at  32  ampere  hours,  its  normal  current  is  4 
amperes;  but  it  can  give  4  amperes  for  but  8  hours, 
and  our  problem  requires  an  11-hour  run.  We  must 
therefore  find  out  what  current  each  cell  can  give 
when  discharged  at  the  11 -hour  rate. 
32  ampere  hours 


11  hours 


=  3  (approx.)  amperes 


PROBLEMS  93 

but  we  can  get  more  than  32  ampere  hours  from  a 
battery  if  it  is  discharged  more  slowly  than  at  the 
8-hour  rate.  Let  us  say  3.25  amperes  is  a  safe  value 
to  assume.  Then  we  are  to  have  three  cells  in  series 
and  enough  groups  to  make  40  amperes. 

-«L-  12.3 

3.25 

Hence  we  shall  need  thirteen  groups  of  cells  in  parallel, 
with  three  cells  in  series  in  each  group. 

It  is  to  be  noted  that  only  in  case  the  voltage 
of  the  circuit  is  that  of  a  single  cell  will  the  sum  of 
the  ampere  hours  of  the  cells  and  the  ampere  hours  of 
the  load  be  the  same.  Ampere  hours  of  cells  in  parallel 
are  additive;  in  series  the  ampere  hours  are  equal  to 
those  of  any  one.  The  energy  expended,  however, 
must  not  be  confused  with  ampere  hours;  energy  = 
ampere  hours  X  volts. 

These  batteries  would  need  to  be  charged  every 
night. 

PROBLEM  3 

What  is  the  cost  of  heating  an  electric  soldering  iron 
which  is  designed  for  a  voltage  of  110  and  has  a  re- 
sistance of  35  ohms?  Energy  costs  5  cents  per  kilo- 
watt hour.  The  iron  is  in  use  8  hours  a  day. 

Solution. 

r      E       110 

C  =  —    =  —    =  3.14  amperes 

R        35 

C2R  =  (3.14)2  X  35  =  345  watts 
345  X  8  hours  =  2,760  watt  hours 

2,760 

X  0.05  =  $0.138,  cost  per  day 

1,000 


94  CHAPTER  X 

PROBLEM  4 

A  portable  electric  drill  has  a  J4  horsepower  motor 
operating  at  110  volts.  If  the  drill  works  at  an  aver- 
age of  five-eighths  full  load  for  5  hours  per  day,  when 
power  costs  4  cents  per  kilowatt  hour,  how  much  does 
it  cost  to  run  the  drill? 
Solution, 

%  horsepower  =  J4  X  746  =  186.5  watts 

186.5  X  %  =  116.5  watts 
116.5  X  5  hours  =  582.5  watt  hours 

582  5 

X  $0.04  =  $0.0233 

1,000 

cost  per  day. 

PROBLEM  5 

A  manufacturer  is  going  to  buy  a  110-volt  direct- 
current  generator  to  supply  the  following:  ten  1 -horse- 
power motors;  sixty  40-watt  lamps;  twenty  100-watt 
lamps;  one  7.5-horsepower  motor;  two  5-horsepower 
motors;  charging  current  for  a  500  ampere-hour 
24-volt  storage  battery. 

What  capacity  generator  would  be  required? 
Solution. 

10  X  746  =  7,460  watts 

60  X  40  =  2,400  watts 

20  X  100  =  2,000  watts 

7.5  X  746  =  5,600  watts 

2  X  5  X  746  =  7,460  watts 

500 

=  62.5  amperes 

normal  charging  current  for  the  battery. 


PROBLEMS  95 

7,460+2,400+2,000+5,600+7,460    =    24,920  watts 

— - —  =  226.54  amperes 
110 

motor  current. 

226.54  +  62.5  =  289  amperes 

It  is  not  likely  that  the  whole  load  would  be  on  at 
any  one  time;  consequently  we  must  assume  some 
load  factor,  to  indicate  the  proportion  of  the  total 
connected  load  which  may  be  expected  to  be  carried 
at  any  time.  Of  course,  if  the  total  load  might  possi- 
bly be  thrown  on  at  once,  no  reduction  from  the  289 
amperes  could  be  taken. 

Assuming  a  load  factor  of  85  percent,  the  maximum 
load  will  be 

289  X  0.85  =  246  amperes 
246  X  110  =  27,000  watts  =  27  kilowatts 
capacity  of  generator  required. 

PROBLEM  6 

A  starting-box  for  a  5-horsepower  220-volt  shunt 
motor  is  to  be  remodeled  for  use  with  a  7. 5-horse- 
power motor  of  the  same  voltage.  What  resistance 
will  be  necessary  in  the  armature  circuit? 

Solution.    A  7. 5-horsepower  motor  takes 

7.5  X  746        watts 

=  =  25.4  amperes 

220  volts 

at  full  load.  Shunt  motor  starting-boxes  are  designed 
to  allow  about  150  percent  of  full-load  current  to 
flow  in  starting.  At  start,  the  motor  has  no  counter 


96  CHAPTER   X 

electromotive  force  (see  Chapter  IV),  and  so  the  re- 
sistance of  the  box  must  limit  the  current  to 

25.4  X  1.5  =  38.1  amperes 
Hence 

220 

R  =  =  5.77  ohms 

38.1 

The  resistance  of  the  armature,  which  would  be 
small,  has  been  neglected. 

There  is  another  point  to  be  considered  before 
undertaking  the  actual  remodeling  of  the  box.  A 
calculation  smilar  to  the  above  gives  the  resistance 
of  the  box  as  it  stands  as  about  8.65  ohms,  with  a 
momentary  current  at  starting  of  25.44  amperes. 
If  the  resistance  coils  already  in  the  box  were  merely 
cut  shorter,  so  that  the  total  resistance  came  down 
to  the  required  5.77  ohms,  the  larger  current  might 
burn  them  out.  Consequently  new  coils  should  be 
supplied  having  larger  current  capacity  than  those 
originally  in  the  box. 

PROBLEM  7 

The  phase  voltage  of  an  alternator  is  127;  the 
windings  are  connected  in  Y  (or  star).  A  three-phase 
induction  motor  is  running  on  current  supplied  by 
this  generator,  its  input  being  3  kilowatts  and  its 
power  factor  85  percent.  What  current  is  flowing  in 
the  line?  (See  Chapter  VI  for  definitions  and  dia- 
grams.) 

Solution.  Using  the  notation  of  Chapter  VI,  we 
have,  in  the  star  system, 


PROBLEMS  97 

EL  =  V/3E  JZ> 
The  line  voltage  then  equals 

EL  =  V/3  127  =  220 

Three-phase  power  (in  either  system)  is  there  stated 
as 

P  =  ^EL  /L  cos  P 
We  have  given 

P  =  3  kilowatts  =  3,000  watts 
EL  =  220  volts 

Cos  B  =  0.85 
.'.  P  =  V/3  220  IL  0.85  =  3,000  watts 

JL  =     ^£22 =  9.27  amperes 

220  X  V/3  X  0.85 

PROBLEM  8 

The  installation  of  problem  5  is  to  be  changed  over 
to  alternating  current.  If  the  power  company  supplies 
2,200-volt  60-cycle  current  at  3%  cents  per  kilowatt 
hour,  what  will  the  motor  equipment  consist  of,  and 
how  much  will  the  power  cost  per  working  day  of  8 
hours,  with  one  battery  charge  every  third  day? 

Solution.  From  a  survey  of  the  list  of  equipment  it 
seems  likely  that  at  least  five  1 -horsepower  motors 
and  one  of  the  5-horsepower  motors  would  require 
variable  speed.  Hence  direct  current  must  be  sup- 
plied, since  there  is  no  good  variable-speed  alternat- 
ing-current motor.  Direct  current  must  also  be  sup- 
plied for  charging  the  storage  battery.  This  would 
probably  be  done  at  night.  The  most  economical 
way  to  charge  the  battery  would  be  by  installing  a 


98  CHAPTER   X 

low-voltage,  motor-driven  generator;  but  since  the 
charge  is  given  but  once  in  three  days  and  then  at  a 
time  when  the  direct-current  generator  is  supplying 
no  other  current,  it  may  be  run  at  reduced  voltage, 
thus  wasting  less  energy  in  regulating  resistance. 

Five  1-horsepower  110-volt  motors  (direct  current) 
require 

5  X  746        „  n 
=33.9  amperes 

110 

One  5-horsepower  110-volt  motor  (direct  current) 
requires  the  same  current.  The  total,  2  X  33.9  = 
67.8  amperes,  is  larger  than  that  required  for  charg- 
ing the  storage  battery;  hence  the  motors  are  the 
determining  factor. 

67.8  X  110  =  7,458  watts 

Thus  if  we  are  buying  "close,"  we  shall  get  a  10-horse- 
power  induction  motor,  220  volts,  driving  a  7.5-kilo- 
watt,  110-volt,  direct-current  generator. 

The  lamps  in  the  plant  may  be  run  from  a  middle 
tap  on  the  secondary  of  one  of  the  transformers,  half 
the  lighting  load  between  the  middle  and  one  line, 
and  half  between  the  middle  and  the  other  line  of  that 
phase. 

That  leaves  five  1-horsepower  motors,  one  of  5 
horsepower,  and  one  7. 5-horsepower  motor,  all  220- 
volt  induction  type. 

60  X  40  =  2,400  watts 
20  X  100  =  2,000  watts  (lamps) 

4  400 

-  =  20  amperes  (current  for  lamps) 


PROBLEMS  99 

Five  1 -horsepower  motors  take  16.95  amperes  (on 
unity  power  factor). 

One  5-horsepower  motor  takes  16.95  amperes. 

One  7. 5-horsepower  motor  takes  25.43  amperes. 

One  10-horsepower  motor  takes  33.9  amperes. 

The  total  alternating-current  motor  current  is 
16.95   +   16.95   +  25.43  +  33.9   =   93.23  amperes 
at  unity  power  factor.     Assuming  that  the  average 
power  factor  of  the  motor  load  is  80  percent,  the  volt- 
ampere  capacity  of  the  transformers  must  be 

en  93 
V  (-     -  +  20)  =  136.54  amperes  X  220  volts  =  30 

0.8       lighting 
motors 

kilovolt  amperes. 

If  the  shop  load  factor  is  75  percent, 

0.75  =  93.23  X  220  X  8  X =    kilowatt  hours 

1,000 

motor  input  per  day  =  123.06. 

Battery  charging  (total)  10-horsepower  motor  run 
at  average  of  half  load  for  10  hours  =  5  X  746  X  10  = 
37.3  kilowatt  hours;  per  day  =  12.43  kilowatt  hours. 

Lamps,  4.4  kilowatts  X  8  =  35.2  kilowatt  hours. 

(123.06  +  35.2  +  12.43)  0.035  =  $5.97  per  day 

motors  lamps  battery 

PROBLEM  9 

A  certain  three-phase  60-cycle  transmission  line 
operating  at  100,000  volts  has  the  following  line  con- 
stants: length,  25  miles;  inductance,  0.119;  capacity, 
0.396  (microfarads);  resistance,  40  ohms.  What 


100  CHAPTER   X 

current  will  flow  at  the  power  house  when  there  is  no 
load  at  any  point  in  the  line? 

Solution.  The  accurate  solution  of  such  problems 
is  difficult;  an  approximation  may  be  obtained  by 
the  simple  method  used  below. 


X2 

E    =  100,000; 
R    =  40; 

X  =  LW—  —  =0.119X27r60 


CW  0.396  XlO"6X27r60 

W  =  2irf 

X  =  0.0067  X  106 
R  is  so  small  compared  with  X  that 


+  X2  =  0.0067  X  106 
very  closely. 

C  =  -  —  -  =  14.93  amperes 
0.0067  X  106 


APPENDIX 

Properties  of  the  Sine-Curve.  Distorted  Waves  of  Current  and 
Voltage — Form  factor;  peak  factor.  Mean  and  Effective 
Values  of  Sine-Curve — Discussion;  calculation.  Torque 
and  Speed  of  Rotation  of  Molecular  Fibers. 


PROPERTIES  OF  THE  SINE-CURVE 

When  a  coil  of  wire  is  revolved  in  a  magnetic  field 
as  shown  in  Fig.  45,  the  velocity  with  which  the  coil 
cuts  the  lines  of  force  is  found  by  resolving  the 
uniform  velocity  of  the  coil  in  its  circular  path  into 
two  rectangular  components,  one  component  parallel 
to  the  lines  of  force.  This  component  has  no  effect  on 
the  coil.  The  component  at  right  angles  to  the  lines 
of  force  is  the  rate  at  which  the  coil  cuts  the  lines  of 
force.  The  e.m.f.  and  current  generated  in  the  coil 
at  any  instant  are  proportional  to  this  velocity  rate. 
This  rate  of  cutting  the  lines  of  force  is  V  sin  0,  in 
which  V  is  the  velocity  of  the  cutting  wires  and  6  is 
the  angle  which  the  plane  of  the  coil  makes  at  any  in- 
stant with  the  horizontal. 

A  curve  drawn  as  in  Fig.  45  is  called  a  sine-curve, 
and  for  the  reasons  above  stated  (see  Chapter  IV) 
such  a  curve  is  used  to  represent  either  the  e.m.f.  or 
the  current  in  an  alternating-current  circuit. 

Let  OA  be  such  a  curve.  Let  it  be  required  to  find 
the  mean  ordinate  of  one  loop  of  the  curve;  that  is, 
the  ordinate  which  multiplied  by  OA  will  give  the 


102 


APPENDIX 


area  of  the  rectangle  OAB C.  If  R  is  the  radius  of  the 
circle,  the  equation  of  the  curve  is  y  =  R  sin  6,  in 
which  0  is  the  variable  angle  R  makes  with  OX. 
Then  representing  the  mean  ordinate  by  EM,  we  have 


FIG.  45 
EM  X  irR  =  Jarea  of  rectangle  =  area  of  sine-curve  = 

=   I    ydx 

But 

dx  =  Rd6 


EM  X 


(-cos 


s: 


2R 


sin  6d8 


Thus  the  area  of  a  sine-curve  is  proportional  to 
the  square  of  its  maximum  ordinate. 

WAVE  DISTORTION 

In  Chapter  IV,  reference  was  made  to  variation  of 
wave  form  in  alternating  current  and  voltage.  To 
give  comparative  values  to  such  distortions,  the  terms 
form  factor  and  peak  factor  have  been  adopted.  The 


APPENDIX  103 

ratio  of  effective  to  average  value  is  called  form  factor ; 
and  the  ratio  of  maximum  to  effective  value  iscalled 
the  peak  factor.  For  the  sine-curve  the  values  are: 

Form  factor  =  — j=.=  1.11  1 

2y2  Isine-curve  only 

Crest  factor  =  )/2  =  1.414  J 

It  is  to  be  noted  that  a  sharply  peaked  wave  will 
be  likely  to  have  a  form  factor  greater  than  1.11,  and  a 
flat-topped  wave  will  in  general  have  a  lower  form 
factor  than  a  sine-wave  of  equal  effective  value. 

MEAN  AND  EFFECTIVE  VALUES — THE  POWER  CURVE 

The  following  is  introduced  to  clear  up  in  the 
student's  mind  any  confusion  into  which  he  may  have 
fallen  concerning  the  use  of  mean  and  effective  values 
of  current  and  voltage. 

Figure  46  is  a  reproduction  of  two  curves,  C  and  L, 
from  Fig.  22,  with  the  addition  of  the  power  curve  Pt 
which  is  obtained  by  multiplying  each  ordinate  of 
the  voltage  curve  L  by  the  corresponding  ordinate  of 
the  current  curve  C. 

If  the  base  be  properly  taken,  the  area  under  curve 
P  will  represent  the  energy  expended  during  the  half 
cycle.  (See  curves,  Figs.  38  and  39.) 

When  the  mean  ordinates  of  curves  C  and  L  are 
known,  the  energy  can  be  obtained  from  the  area  of  L 
as  follows: 

Power  =  effective  volts  X  effective  amperes 
effective  volts  _  effective  amperes 

~     ^^    J.  •  X  JL  |  ™~       1 .  1  i 

mean  volts  mean  amperes 


104 


APPENDIX 


Hence 

Power  =  mean  volts  X  mean  amperes  X  (1.1 1)2 
and  the  energy  per  half  cycle,  when  the  mean  current 
is  unity,  is 

Energy  =  area  of  voltage  curve  X  1.2321 


FIG.  46 


APPENDIX  105 

CALCULATION   OF   MEAN   AND    EFFECTIVE  VALUES 

Distinction  between  mean  and  effective  values  of 
sine-wave  functions:  Alternating-current  machinery 
is  designed  to  operate  as  nearly  as  possible  on  pure 
sine-wave  forms  of  voltage,  current,  and  magnetic 
flux  variations,  calculations  and  predeterminations 
being  thereby  made  more  easily. 

It  should  be  noted  that  voltmeters  and  ammeters 
do  not  read  the  geometric  mean  of  voltage  or  current, 
but  a  value  slightly  higher,  called  the  "effective" 
value. 

If  i  =  sin  6  is  the  equation  of  an  alternating  current, 
i*R  is  the  instantaneous  power,  and  substituting 
P  sin2  0  for  i2,  we  have 

PR  =  P  sin2  OR 

Let  us  find  an  expression  for  the  average  power, 
which  can  be  done  by  integrating  the  expression 

-  between  the  limits  of  0  and  T;  that  is,  over 

one  loop  of  the  curve  of  i  —  I  sin  0 

rirPsinz0R    ,a         PR   Cv     .  2 
P   =          -  d0  =   -          sin*  0d0   = 

J  TT  TT      J     o 


-T  (  r  *  -\  P"" 

27T  \J  o  2  J  o 


PR  ,         /O       0\      PR 


106  APPENDIX 

Now  from  this  expression  for  average  power  we  can 
obtain  the  value  of  alternating  current  which  is  the 
equivalent  of  some  direct  current  which  would  give 
equal  power,  and  this  value  of  alternating  current  is 
called  the  effective  or  virtual  value  of  the  current, 
and  is  the  value  read  by  alternating-current  meters. 

PR       /  /  V 
Average  power  =    —    =  j^gjj   R 

whence  it  is  seen  that  —r-  is  the  value  which  cor- 
responds to  the  direct  current  which  will  have  the 
same  heating  effect  and  provide  the  same  power  as 
the  alternating  current  which  has  /  for  its  maximum. 
When  an  engineer  refers  to  an  alternating  current  of 
10  amperes,  he  refers  to  this  meter  value  or  effective 
value. 

Whence  the  effective  value  and  maximum  values 
are  related  thus: 

leff.      =      J~£=      0.707     Imax, 

The  effective  current  value  is  slightly  higher  than 
the  average  ordinate  of  the  corresponding  sine-curve, 
as  will  be  seen  from  the  following  integration: 

/*  1      /  /v 

/  sin  BdB  X  -  =  -    1      si 

0  7T  7T  J    O 


sn 

o 


Whence 


2 
iave.    =  -  X  Imax.  =  0.636  Imax. 

TT 


APPENDIX  107 

Whence  it  appears  that  the  average  current  is 
0.636  of  the  maximum,  while  the  effective  is  0.707 
times  the  maximum. 

TORQUE  AND  SPEED  OF  ROTATION  OF 
MOLECULAR  FIBERS 

While  it  serves  no  immediately  useful  purpose,  it 
may  be  interesting  to  examine  the  equation  P  = 
M  X  A  in  an  attempt  to  assign  possible  values  to  the 
torque  and  angular  velocity  of  individual  molecular 
fibers. 

The  diameter  of  the  elementary  fiber  can  hardly  be 
less  than  that  of  a  single  molecule,  though  it  may  be 
greater.  From  the  table  of  molecular  dimensions 
given  by  the  Smithsonian  Physical  Tables,  it  may  be 
assumed  that  4  X  lO'8""  (=  1.575  X  10'8  inches  = 
0.00000001575  inches)  is  an  index  of  the  order  of 
magnitude  of  the  copper  molecule. 

Intermolecular  interstices  seem  always  to  be 
characteristic  of  matter.  If  we  assume  that  80  per- 
cent of  the  apparent  area  of  a  certain  wire  is  composed 
of  cross-sections  of  molecular  fibers,  there  will  be  in 
the  wire  a  total  of  F  fibers,  where 

(1)  F  =  °-** 

s 

S  =  area  of  wire,  and  5  =  area  of  filament. 

For  purposes  of  calculation  let  us  use  a  copper  wire 
having  a  diameter  of  0.5  inch,  and  let  us  assume  that 
it  is  a  part  of  a  line  which  is  transmitting  800  horse- 
power in  a  1,200- volt  system.  This  would  correspond 
to  a  feeder  wire  of  an  interurban  electric  railway. 


108  APPENDIX 

The  area  of  the  wire  is 

(2)      5  =  irR2  =  7r(0.25)2  =  0.1962  square  inch 
The  area  of  a  filament  is 
.575    X    I 


_  ^  _  ^1. 


2 

=  1.947  X  10'16  square  inches 
Hence  the  number  of  molecular  fibers  is 

(4)  F  =  °-8  X  °'1%126  =  0.0807  X 
1.947  X  10~16 

=  807,000,000,000,000 

The  assumed  rate  of  transmission  was  800  horse- 
power, or 

800  X  33,000  =  26,400,000  foot  pounds  per  minute 
Therefore  the  rate  per  fiber  is 

26,400,000  =  264  X  105 

807,000,000,000,000  "  807  X  1012 

=  0.3272  X  10"7 

=  0.00000003272  foot  pounds  per  minute  per  fiber 
This  represents  P  in  the  equation 

(6)  P  =  MA 
whence 

(7)  P  =  0.3272  X  10 7    =  torque  per  fiber   X    2*N 
It  will  be  seen  that  an  arbitrary  value  must  now  be 

assigned  to  torque  or  revolutions,  and  the  difficulty 
is  presented  of  attempting  to  find  a  reason  for  making 
whatever  assumption  may  be  decided  upon.  For  it 
is  evident  that  if  the  problem  be  set  before  ten  indi- 
viduals, there  might  be  presented  ten  different  solu- 
tions from  which  it  would  be  impossible  to  choose  one 
as  being  better  than  the  others.  This  fact  is  one 


APPENDIX  109 

more  point  of  evidence  in  support  of  the  rotational 
theory  of  electrical  transmission. 

The  current  flowing  in  this  wire  may  be  found 
from  the  conditions  previously  stated.  Eight  hundred 
horsepower  is  being  transmitted  at  1,200  volts.  There 
are  746  watts  in  one  horsepower,  hence 

W  =  800  X  746  ==  596,800  watts 
=  W  _  59MOO  _ 
E         1,200 

If  we  examine  Equation  6,  we  find  that  the  right- 
hand  member  consists  of  two  quantities  which  may  be 
expressed  thus: 

M  -  total  torque  per  fiber  =  torque  per  volt  X 
number  of  volts; 

A  =  2irN  =  angular  velocity,  where  N  =  r.p.m. 
per  ampere  X  number  of  amperes. 

In  the  present  example 

(9)  P  =  0.3272  X  10'7  foot  pounds  per  minute  (from 
Equation  5) 

(10)  M  =  1,200  T 
where 

1,200  =  number  of  volts; 

T  =  torque  per  volt,  per  fiber. 

(11)  A  =  2wN  =  27r(497.33w) 
where 

497.33  =  number  of  amperes; 
n  =  r.p.m.  per  ampere. 

Let  us  assume  for  purposes  of  calculation  that  n  = 
104  =  10,000.  A,  then,  =  27r(497.33  X  10,000)  = 
31,200,000;  or  A  =  31.2  X  106. 


110  APPENDIX 

(12)  P   =   M  X  A 

0.3272  X  lO'7  =  1,200  T  X  31.2  X  106 

(13)  T  =  87.4  X  10'20  foot  pounds  per  fiber,  per  volt 
or  expressed  decimally, 

T  =  0.000000000000000000874  foot  pounds  per  fiber, 

per  volt 
when  n  =  10,000  r.p.m. 

T  has  been  reduced  to  torque  per  volt  in  order  to 
see  what  the  total  torque  would  be  in  other  cases  than 
our  hypothetical  800-horsepower  transmission,  which 
involved 

r.p.m.  =  4,973,300 
and 

torque  per  fiber  =  1,200  T  =  0.0000000000000010488 
foot  pounds 

If  the  voltage  were  raised  to  200,000,  which  is  the 
highest  practically  employed,  the  torque  per  fiber 
would  be  only  0.0000000000001748  foot  pounds  and 
hence  well  within  the  bounds  of  possibility. 


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